When and how is the advocacy of violence illegal? Today I saw a t-shirt reading "Kill a liberal, save a taxpayer." Also I have heard that the right to free speech does not extend to the right to shout "fire!" in a public theater. But if I did choose to exercise non-free speech, what would happen? What statutes exist against wearing a t-shirt that advocates violence against a particular group? If I wore such a t-shirt and the police decided it broke the law, what would be the consequence? Is this illegal at the federal, state, or local level? How does it vary, if not enforced by the US federal government?
There are venue restrictions where political speech is restricted, such as on military bases; content restrictions (transmitting classified information to the world); you cannot defraud by saying false things in order to get something, you cannot defame a person, you cannot speak obscenely (though it's hard to tell what counts as "obscenity"). You cannot appropriate other people's property in speaking (i.e. copyright law is a restriction on speech). The type of speech restrictions seem to pertain to speech and violence caused by such speech. A classic limitation is that you cannot speak "fighting words" (Chaplinsky v. New Hampshire 315 U.S. 568), which in 1942 meant calling someone a "damned racketeer" and "damned Fascist", which the court characterized as "inherently likely to provoke a violent reaction". The court subsequently refined its position on "provocative" speech. In Virginia v. Black 538 U.S. 343 a law against cross-burning was found to run afoul of the First Amendment as a restriction on political expression, but it would be fully consistent with The Constitution to outlaw "cross burning carried out with the intent to intimidate". This states may "prohibit only those forms of intimidation that are most likely to inspire fear of bodily harm". The current position is that you cannot incite to the imminent use of force. In Brandenburg v. Ohio 395 U.S. 444, the court stated that the First Amendment does not "permit a State to forbid or proscribe advocacy of the use of force or of law violation except where such advocacy is directed to inciting or producing imminent lawless action and is likely to incite or produce such action". There are myriad laws against threats, for instance in Washington you may not "knowingly threaten(s) to cause bodily injury immediately or in the future to the person threatened or to any other person", and you can't do that ("knowingly cause another to believe that the offender will cause serious physical harm to the person or property of the other person") in Ohio either. You can't get away with threatening "to commit a crime which will result in death or great bodily injury to another person" in California. This class of restrictions on speech seems to be quite robust. You may not induce panic in Ohio, e.g. shout "fire" in a theater -- I don't know if any other state has such a law.
As mentioned in a comment by @Dancrumb, the exact policies of each local police department will be different, and there are thousands of them. There is a relevant requirement at the Federal level according the Department of Justice, but it is not clear to me to what extent this applies to peers and not just supervisors: An officer who purposefully allows a fellow officer to violate a victim's Constitutional rights may be prosecuted for failure to intervene to stop the Constitutional violation. To prosecute such an officer, the government must show that the defendant officer was aware of the Constitutional violation, had an opportunity to intervene, and chose not to do so. This charge is often appropriate for supervisory officers who observe uses of excessive force without stopping them, or who actively encourage uses of excessive force but do not directly participate in them.
"If it were not assize-time, I would not take such language from you." (said while grabbing the handle of sword) This is a famous conditional threat where the speaker/actor was not found to express intent to do harm; perhaps better called a negative condition. This probably confuses matters but if you are to search for more answers this could be a good place to start. One of the elements of common law assault is that the threat must be able to be carried out immediately; it must be imminent. I do not have a cite for this but I recall that this means that conditional threats are excluded from assault. So calling a politician on the phone and telling them that if they do not drop out of a race you will hurt them is not assault. So, "You cut that out now or you’ll go home in an ambulance" sounds a lot like, "stop or you will get hurt." The victim has the opportunity to avoid the danger; the threat is not imminent. But the facts here are interesting because the speaker touched the victim while speaking which might mean fear of imminent was real. But they were in a crowded room in front of cameras - could the victim really feel that threat was imminent? Plus, the "you will go home" implies a future harm. Oh, and the speaker does not say "I will hurt you," maybe she was actually trying to protect the victim from someone else's actions. Like when my teacher knew someone was waiting outside the classroom to fight me and she told me, "if you go out there you will get hurt!" I would hope that a jury would consider this hard bargaining.
Questions about "why a law is ..." are political questions not legal questions and you may get better traction on politics. However, I will address the legal issues and offer some speculation on the politics. The states named in the preamble to the Constitution (an Act of British Parliament) as original states were New South Wales, Victoria, Queensland, South Australia and Tasmania. Western Australia was not named at the time of the passing of the Act or Royal Assent because the people of that colony had not vet made their mind up. Legal Issues WA decided to join in a vote held on 31 December 1900 and Australia came into being on 1 January 1901. Therefore, even though not named as such, WA was an "original state". Since all 6 states in the Federation are "original states" the clauses have no practical effect at present. However, there have been a number of proposals to add new states, either by subdividing existing states or by granting statehood to the territories of Northern Territory and/or the Australian Capital Territory. If such were to come to pass, the clauses would have practical effect. In 1998, Norther Territorials rejected an offer of statehood that would have given them 3 senators as a state and 2 representatives based on population (currently they have 2 senators and 2 representatives). Clearly, they were not being given the same privileges as an "original state". In 2015 all Australian governments agreed in principle that the NT should become a state by 2018, however, as it is now 2017 and no action has been taken this seems unlikely. Political Issues Politics is complicated: just as much in the late 19th century as it is in the early 21st. Negotiations between the colonies were fraught and federation was by no means a certain outcome. New Zealand and Fiji dropped out early and each forged its own path to nationhood. However, by the late 1890s it was clear that the 5 eastern colonies would federate with or without Western Australia. It seems likely that this provision served multiple purposes including: putting pressure on WA to join at the outset - the deal they got as a "Johnny come lately" may not have been as good. protecting "white" Australia - the drafters of the Constitution were men of their times, that is to say: racist, misogynist bigots. Any non-original states were likely to be former British colonies in the Pacific or South-East Asia, this clause would allow the nation to reduce the influence these non-white states might have.
It could lead to investigation and wasting of government resources, but it is not a crime and not civilly actionable, under the circumstances described. The First Amendment right to say crazy things is pretty broad.
The First Amendment generally prohibits the government from taking any actions to limit your speech, the same as adults. There are certain exceptions, particularly if you are enrolled in a public school, which has some latitude to impose speech restrictions to “avoid substantial interference with school discipline or the rights of others” Tinker v. Des Moines Independent Community School Dist., 393 U.S. 503, (1969). While you are a minor, though, your parents have virtually unlimited freedom to limit your freedom of speech. If they want to punish you for buying a hat or saying things they disagree with, they can generally do that. EDIT: The hat you've linked to probably would not fall within the Tinker exception. A similar case arose in Schoenecker v. Koopman, 349 F. Supp. 3d 745 (E.D. Wis. 2018), where a student was removed from class for wearing similar apparel supporting Second Amendment rights. The school made a vague allegation that his shirts were disruptive, but the court found that its concerns about disruptions were largely unreasonable and unsubstantiated. See also N.J. ex rel. Jacob v. Sonnabend, No. 20-C-227, (E.D. Wis. Nov. 6, 2020).
An example of where this is not allowed is Seattle, WA. Municipal code SMC 12A.06.025 states It is unlawful for any person to intentionally fight with another person in a public place and thereby create a substantial risk of: Injury to a person who is not actively participating in the fight; or Damage to the property of a person who is not actively participating in the fight. B. In any prosecution under subsection A of this Section 12A.06.025, it is an affirmative defense that: The fight was duly licensed or authorized by law; or The person was acting in self-defense. You can see from adjacent sections that "mutual combat" is not legal. I recognize that there is this meme about Seattle, but this is a distortion of an incident when the police turned a blind eye to a fight. We have police issues, no doubt: there is nothing legal about such fights. Of course, for a licensed event, you can "fight". Of course the potential legality depends on how mutual combat is defined. Illinois v. Austin 133 Ill.2d 118 and citations therein, subsequently Illinois v. Thompson, 821 NE 2d 664 define it thus: Mutual combat is a fight or struggle which both parties enter willingly or where two persons, upon a sudden quarrel and in hot blood, mutually fight upon equal terms and where death results from the combat. Similar death-definitions are found in Donaldson v. State, 289 SE 2d 242, Iowa v. Spates, 779 NW 2d 770. The law looks askance of such behavior. For the sake of clarity, a term other than "mutual combat" would be preferable.
Leonard's law says that the school can restrict speech if it is against the religious tenets of the organization. Now I don't think school uniforms fall are part of the tenets of Catholicism, so why can private schools enforce dress codes? I see there's a misunderstanding of the Leonard's law. But I fault the California legislators (not you) for that confusion, since the statutory language is ambiguous and leads to the reasonable interpretation your inquiry reflects. The statute reads: (c) This section does not apply to a private postsecondary educational institution that is controlled by a religious organization, to the extent that the application of this section would not be consistent with the religious tenets of the organization. It is not that a religious school is allowed to restrict speech if it contravenes the tenets of that religion, but that the statute altogether is inapplicable to religious schools because it is considered an infringement of fundamental liberties to which religious institutions are entitled in the US. The [religious] school would prevail on the basis of the ecclesiastical abstention doctrine. See Dermody v. Presbyterian Church (U.S.A.), 530 S.W.3d 467, 474 (2017): The ecclesiastical-abstention doctrine prohibits courts from deciding cases "dependent on the question of doctrine, discipline, ecclesiastical law, rule, or custom, or church government[.]". (Please note that I strongly disagree with the application of that doctrine in the Dermody case and I consider it impermissibly outdated for the controversy litigated therein, but that is a separate issue). A religious school could convincingly argue that the judicial review of its uniforms policy infringes matters of ecclesiastical discipline/rule/custom, aspects which ultimately "involve an internal church dispute over religious authority or dogma" Roman Cath. Archbishop of LA v. Super. Ct., 32 Cal.Rptr.3d 209, 220 (2005). Infringements of ecclesiastical abstention and akin doctrines would be outweighed only in "compelling" [cases] because "the duty to prosecute persons who commit serious crimes is part and parcel of the government's `paramount responsibility for the general safety and welfare of all its citizens'" Roman Cath. Archbishop of LA v. Super. Ct. at 225 (brackets added in this answer). A free speech controversy such as the school's uniform policy simply does not meet that threshold.