I'm not a lawyer, but if I recall correctly, you have the right to not say anything except 'I want to consult my lawyer' when questioned by police. If someone does not remain silent or wait for their lawyer, but talks to police and answers their questions, I don't think that's necessarily unconstitutional. Of course, the police can't use torture or anything, but they can trick people by, for example, triggering their guilty conscience.
Secondly, I'm pretty sure they don't have to get a warrant in some circumstances, such as looking for a missing person who was last seen in that location, or being told by someone that they'd witnessed a murder there. They probably can't remove evidence, but they can go in and look for obvious signs of the person, a body or crime scene.
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