Is lying at all in a testimony considered perjury? Say Bob was murdered at a coffee shop, and Jerry was a witness at the scene. At the trial, he is under oath, and gives his testimony. In his testimony, he explicitly claims that he was watching America's Funniest Home Videos up to the point when the murderer entered the shop. Later in the trial, video evidence, while first and foremost showing that the defendant did indeed murder Bob, also showed that Jerry was watching something that wasn't America's Funniest Home Videos at the point when the murderer entered the shop. For the purpose of simplicity, let's assume that what he was watching is perfectly fine legally speaking. Let's also assume that he knowingly lied about what he was watching, and didn't make a simple mistake. Even though this detail was virtually irrelevant to his testimony, he still technically lied under oath, which is the definition of perjury. Could this result in Jerry being criminally prosecuted and/or his entire testimony becoming invalid?
Take the US federal perjury statute as an example: Whoever having taken an oath before a competent tribunal, officer, or person, in any case in which a law of the United States authorizes an oath to be administered, that he will testify, declare, depose, or certify truly, or that any written testimony, declaration, deposition, or certificate by him subscribed, is true, willfully and contrary to such oath states or subscribes any material matter which he does not believe to be true ... The key word is material, meaning relevant, important, essential. From your description, the specific show that Jerry was watching is not material to the proceeding in which the testimony was given, so Jerry has not perjured himself. Other common law jurisdictions will likewise have materiality as an element of perjury.
Gabbard accusing Harris of obstruction doesn't make Harris guilty of obstruction. There is no legal progression between a statement by one party about the existence or veracity of a fact that could prove a crime by another party, and that other party and the veracity of the facts or the crime itself and forgone guilt, other than the statement by the first party itself could possibly be defamatory by publishing (making public statements) false facts. By saying "If taken at face value (not saying it is actually true)", I think you're confusing obstruction with defamation. If Harris did factually "block evidence", and it rises to the threshold of obstruction as determined by a prosecutor, then Gabbard isn't stating false facts and is not defaming Harris. If Harris did not factually "block evidence", as determined by documents and evidence, then Gabbard could be stating false facts, but both are public figures, so the threshold of defamation is higher in that it requires actual malice by Gabbard knowing her statement is false or by reckless disregard for the truth. To go a step further, to prosecute Harris, a prosecutor has to determine if Harris obstructed according to evidence, and charge her. Harris could have obstructed and not be charged for either political reasons or reasons regarding evidence, but that's a different question.
As a comment by @DavidSchwartz notes, this is not wrong. Questions of law but not fact are allowed. It is worth noting that the line drawn is arbitrary. In Colorado, where I practice, jurors issue written questions (pre-reviewed by the judge and counsel for all parties before being presented) to witnesses at the close of the testimony of each witness called by a party to testify. This is very helpful to counsel, as it provides indirect evidence of whether the jury understands what they are being told, and often juries will directly ask questions that for tactical reasons, both parties have refrained from asking that go to the heart of the matter. It also frequently clarifies misunderstandings that trained legal professionals assumed were not made about terminology. This is more problematic in criminal trials, where jury questions could provide evidence pushing a case over the threshold of proof needed to prove beyond a reasonable doubt that the prosecution failed to provide, than in civil cases with a preponderance of the evidence standard. Also, as a matter of reality, when jurors ask questions, counsel often loathe to object even when they have valid grounds to do so, for fear of offending the decision-maker, unless it is really critical to keep certain information away from the jury.
Your belief in the truth of a statement is of only limited value in a defamation suit. Saying that a person is incompetent in that person's profession can certainly be defamation, and may even be defamation per se, depending on the jurisdiction. If the person sues and the speaker asserts truth as a defense, the speaker may have to prove that the statement is true. Why would one need to say anything at all in such a situation? Whether such a statement was a factual statement or a statement of opinion would depend very much on the detailed circumstances. But it is at least possible that a suit might be filed and won on this fact pattern. See this answer for more on defamation under US law.
Truth is a defense to defamation Bob must prove the truth of his statement if Rob sues - there is a reverse onus for this defense. Because this is a civil trial the burden is balance of probabilities. Provided Bob can prove Rob stole his bike he will win. A conviction for doing so is pretty good (but not necessarily conclusive) evidence. Absent that, Bob would need other evidence. Of course, if Bob has said that Rob was convicted of stealing the bike, he’s going to lose.
False statements are generally protected by the First Amendment. If the video was an obvious gag or work of fiction, in which a reasonable person would understand that you were not truly endorsed, your false statement would almost certainly be protected by the First Amendment. But many false statements are not protected, typically because of their negative effects on other parties. Classic examples include defamation, perjury, and fraud. A lie about an endorsement could easily fall on either side of that line. If you are making the video in a context in which reasonable viewers would be led to believe that you truly were endorsed by that company, I could easily imagine a set of circumstances that would lead to liability: If you're releasing the video as part of a commercial endeavor, you're looking at possible claims for false advertising, unfair competition, and trademark violations (false designation, dilution, etc.). If your video is harmful to the company's reputation, you're facing a potential claim for product disparagement or trade libel. These are the corporate equivalents of the libel actions most people are more familiar with, and they generally require proving the same kinds of facts. A right of publicity/misappropriation of identity action is possible but very unlikely. Most states are very stingy about allowing businesses to bring these claims, so this one would probably be very jurisdiction-specific. Proving the case generally requires the company to show (1) that you made a commercial use of their identity, (2) that they did not authorize it, and (3) that the use caused economic harm. Beyond those, there's a whole host of business torts that I could try to squeeze this into but that are probably less likely given the hypothetical you've presented. I'd also note that your assumption about proving damages is not well-founded. Especially among the largest companies, businesses are very protective of their brands, and they invest absurd amounts of money to protect them. Pick any company with a high enough profile that someone would want to fake their endorsement, and I'd be willing to bet that their legal department could pretty quickly prepare a detailed estimate of the related damage to their brand. If a case went to trial, you'd normally expect to see an economist present an expert report calculating damages, which would begin (but not end) with any unjust enrichment you experienced because people thought you had the company's endorsement.
united-states The Sixth Amendment gives a defendant in a criminal case the right to “confront one’s accuser”, and the Supreme Court has taken a notably originalist view of this right, holding that this means face-to-face cross examination under virtually all circumstances. While there are some small exceptions related to minors and to witnesses who became unavoidably absent after giving a sworn statement (none of which could apply to Alice), and while the Supreme Court hasn’t explicitly ruled out cross examination over videoconference, the idea of anonymous testimony in a criminal case is unthinkable. Part of an effective cross examination is arguing why the witness’s testimony might be unreliable, and a defendant who didn’t know whose testimony it was would be hamstrung at that.
You misunderstand the significance of the phrase "innocent until proven guilty." This is in part because you are not considering the entire phrase. The full phrase is that an accused party is "presumed innocent until proven guilty." This does not mean that the accused is innocent, only that criminal procedure must take as its starting point that the accused did not commit the crime. The major implication of the presumption, and indeed its original purpose, is that it places the burden of proof on the prosecution. This means that if a prosecutor asserts that you stole something, you do not have to prove that you did not. Rather, the prosecutor must prove that you did. The only reason to present evidence of your own is to rebut the prosecutor's evidence. Another practical implication is that a decision to detain someone awaiting a criminal trial may not be based on the assumption that the accused committed the crime. On the other hand, that decision is not based on the assumption that the accused did not commit the crime. There is a presumption of innocence, but no assumption of innocence, and the government is not obliged before the person is convicted to treat the person as if there is no accusation or charge. Wikipedia has a decent discussion. If we modify your question accordingly, it becomes How can two people be presumed innocent until proven guilty if their stories conflict? Now the answer should be clear. The prosecutor must develop evidence that shows which one of the people has committed the crime. If the prosecutor cannot do that, neither person may be punished.