Trespass and the police Can a US cop ask a land owner if a person is trespassing if no complaint has been made? i.e. can they ask a shop owner if a person is trespassing in their car park or outside the shop?
Yes, of course. Anyone can walk up to a shop owner and ask them pretty much whatever questions they want; police have just as much of a right to do that as anyone else. The shop owner doesn't have to answer, but police are free to ask.
There's the question whether something is lost property or abandoned property. You'd be allowed to keep abandoned property, but keeping lost property without looking for the owner is in many places considered theft. A car on your land is quite likely abandoned by the last driver (people don't usually lose cars). But the question is whether it is abandoned by the owner; if the car looks like it has some value then it is unlikely to be abandoned by the owner and more likely that it has been stolen. I'd report the car to the police; then it's up to them to find the owner or not. If they can find him, and the car was not abandoned, but actually lost (unlikely) or stolen (more likely), you have the satisfaction of being an honest person helping either a very stupid car owner or a crime victim to get their property back. If they can't find him, usually the property will then belong to the finder.
Is there a law requiring a landlord to respond to rental agencies request for information in regards to previous tenants? No. There is no such law.
Indirectly, no the wording of the caution is "You do not have to say anything, but it may harm your defence if you do not mention when questioned something which you later rely on in court. Anything you do say may be given in evidence." It is illegal to question someone if they have asked for legal advice. so if you ask for a solicitor they are not allowed to question you, and you cannot (by definition) fail to answer their questions if they're not allowed to ask them. Reference from https://www.gov.uk/arrested-your-rights/legal-advice-at-the-police-station "Once you’ve asked for legal advice, the police can’t question you until you’ve got it - with some exceptions."
You cannot booby trap your property, see Can I booby-trap my property against police? You can, however, protect your property in any reasonable way that will not cause harm to innocent trespassers. So things like fences with obvious barbed wire, road spikes that would damage tires but not people etc. From the images it seems like the nature of the spikes are such that they would injure someone who stepped on or fell on them and probably cross the line into booby trap territory.
The 5th amendment protects you from self-incrimination. If by possessing a firearm you are in violation of the law you cannot be compelled by law to reveal this information. If the police discover you have a gun in violation of the law you can be arrested and prosecuted for that offense. They cannot additionally prosecute you for not telling them about a gun. I thought there was a supreme court case about this specific situation, but the closest I can find for now is Haynes v. United States. The 7-1 majority held that people prohibited from possessing firearms cannot be compelled to register their firearms that they are possessing illegally. They are stopped by the police and asked this question. They answer truthfully. Then they can be arrested and prosecuted for the illegal possession of the gun. How does this square with the right not to self-incriminate? Or is asking the question considered to be a search? Police can generally ask whatever they want. If you choose to waive your 5th amendment rights, that's your mistake. Can the state prosecute this person for carrying the illegal weapon? The state can generally prosecute crimes it knows about. So yes, in this case they can. Suppose that next to the weapon a stash of illegal drugs is discovered, which was only found due to the action taken to secure the weapon. Can the state prosecute for that? The state can generally prosecute additional crimes it uncovers during investigations or other lawful actions. So yes, this can be prosecuted.
Let's put to bed the myth of privacy that is at the heart of your question: in R v Sotheren (2001) NSWSC 204 Justice Dowd said “A person, in our society, does not have a right not to be photographed" So they can ask you to stop; its bad manners if you don't but it is not illegal. If they are the controller of the property then they can stop you filming from their property but they cannot stop you filming into their property from outside (either public land or land where you do have permission). See How do laws affect photography of non-humans in public when people may be in the frame?
No The criteria for adverse possession is that you have to be in possession without permission. A tenant, even one that pays no rent (or stops paying rent), has permission.