Exclusionary Rule and testimony Let's say a police officer searches a property without a warrant, and discovers a couple of people imprisoned there. Would their testimony against the owner later be inadmissible as it's a result of the illegal search?
First, if the police officer had reasonable cause to believe that a crime was in progress then the search would not be illegal in the first place. However, let's assume the search was illegal. Normally the evidence would be excluded under the exclusionary rule. However, there are two exceptions known as the independent source and attenuation doctrines. The evidence of the police officer as to the imprisonment would be excluded. However, the evidence of the victims is independent of that illegal search and their testimony would be admissible. Further, that testimony would allow independent discovery and admissibility of any physical evidence in the property. However, if the victims were dead, then there would be no independent discovery and none of the evidence would be admissible.
You are correct. A judge may only issue a warrant when it is supported by an affidavit, in which the officer seeking the warrant swears under oath to the facts supporting the warrant. Lying on the affidavit would constitute perjury. But judges very frequently just rubber-stamp the warrants without meaningfully reviewing the affidavits, so the primary form of oversight would be the defendant's Fourth Amendment challenge asserting that the warrant wasn't supported by probable cause. If a judge does review the warrant application and finds the officer's statements not to be credible, he can refuse to sign the warrant, and he is free to also carry that credibility determination to subsequent warrants sought by the same officer or other officers in his department.
Criminal charges are filed and prosecuted by the government, on behalf of the public, and there is no requirement for approval or cooperation by the victim. As a policy matter, a DA may decide to not charge a person in case the victim is unwilling (though less so in cases of domestic violence), perhaps because of the widespread impression that the victim has to "press charges" (which indeed they would have to do in the case of private prosecution, which is no longer allowed in the US). The alleged victim's reddit exchanges on the topic are here. An uncooperative victim does not make a good witness, even if they are compelled to testify. The police statement that they cannot pursue an investigation should not be interpreted as a statement of law, it's probably a statement of policy and practicality.
An existing law actually prohibits using census data "against" a person, see this recent question. The 5th Amendment ("nor shall be compelled in any criminal case to be a witness against himself") is not interpreted to imply an absolute privilege to not answer, it means that your answer cannot be used against you in a criminal case. You can be compelled to testify "against yourself" if you are granted immunity from prosecution.
Police officers are authorized to use force regardless of what they are wearing, to effect an arrest. One issue will be whether the defendants should know that they were under arrest, but there is no requirement to utter particular phrases when dealing with a combative lawbreaker. There will be an internal investigation at some level to determine whether the officers violated any department policy, and no doubt the video and testimony of those in the are would be relevant. There probably is some policy to the effect that you have to distinguish yourself from a street vigilante (you have to state your authority), though I can't find any specific online publicly-available department rules. There is no law that requires an officer to say that he is one, or to show his badge, before starting an arrest.
Yes and No See Katz v. United States, 389 U.S. 347, 351 (1967): What a person knowingly exposes to the public, even in his home or office, is not a subject of Fourth Amendment protection. But what he seeks to preserve as private, even in an area accessible to the public, may be constitutionally protected. The glass has clearly been "knowingly exposed" and if that glass was, on the normal methods of examination by the officer (sight, smell etc.), evidence of a crime then it would be able to be used as evidence. However, any DNA on that glass is not "knowingly exposed" - finding the DNA requires a laboratory, specialised equipment and trained technicians.
Let's say you leave fake drugs in someone's yard. Eg, you expect the police to be dumb enough to believe it and arrest the other person Obstructing a public officer, specifically "deliberately hindering a public officer from carrying out official duties". Trespassing.
No That is nothing but fiction. Assuming that this is in the US, the police would (probably, there are some exceptions) have had to deliver the well-known "Miranda" warnings, that the suspect has the right to silence, the right to consult a lawyer, and the right to have a free lawyer if unable to afford one, and that statements may be used against the subject. If, after those warnings, the suspect chooses to confess, or to make a statement, that confession or statement would be fully admissible, even if the suspect did not have a lawyer present, unless there was some other reason for the statement to be excluded. No such reason is mentioned in the question. It is simply not the case in the US that a confession is excluded just because no lawyer was present, nor is that the law anywhere that I know of. If the police failed to give the warnings when they were required, then any statements or confessions would be excluded. The decision in Miranda v. Arizona, 384 U.S. 436 (1966) says: law enforcement officials took the defendant into custody and interrogated him in a police station for the purpose of obtaining a confession. The police did not effectively advise him of his right to remain silent or of his right to consult with his attorney. Rather, they confronted him with an alleged accomplice who accused him of having perpetrated a murder. When the defendant denied the accusation and said "I didn't shoot Manuel, you did it," they handcuffed him and took him to an interrogation room. There, while handcuffed and standing, he was questioned for four hours until he confessed. During this interrogation, the police denied his request to speak to his attorney, and they prevented his retained attorney, who had come to the police station, from consulting with him. At his trial, the State, over his objection, introduced the confession against him. We held that the statements thus made were constitutionally inadmissible. ... the prosecution may not use statements, whether exculpatory or inculpatory, stemming from custodial interrogation of the defendant unless it demonstrates the use of procedural safeguards effective to secure the privilege against self-incrimination. By custodial interrogation, we mean questioning initiated by law enforcement officers after a person has been taken into custody or otherwise deprived of his freedom of action in any significant way. [Footnote 4] As for the procedural safeguards to be employed, unless other fully effective means are devised to inform accused persons of their right of silence and to assure a continuous opportunity to exercise it, the following measures are required. Prior to any questioning, the person must be warned that he has a right to remain silent, that any statement he does make may be used as evidence against him, and that he has a right to the presence of an attorney, either retained or appointed. The defendant may waive effectuation of these rights, provided the waiver is made voluntarily, knowingly and intelligently. If, however, he indicates in any manner and at any stage of the process that he wishes to consult with an attorney before speaking, there can be no questioning. Likewise, if the individual is alone and indicates in any manner that he does not wish to be interrogated, the police may not question him. The mere fact that he may have answered some questions or volunteered some statements on his own does not deprive him of the right to refrain from answering any further inquiries until he has consulted with an attorney and thereafter consents to be questioned. That tells you exactly what the police are forbidden to do. Nowhere does it say that a lawyer must be present. Indeed it says the opposite: The defendant may waive effectuation of these rights, provided the waiver is made voluntarily, knowingly and intelligently. That means that s/he can confess after being warned, and such confession would be admissible, provided that s/he knew and understood those rights.