Look, I’ve been told in the past that it’s not a good idea to talk to the police without a lawyer present. The people who told me this were experts on this sort of thing and the point they made is that innocent people have been convicted by accidentally saying something that implicated them. They also made the point that this has happened even when the police officers themselves believed the person was innocent because those officers turned the case over to a District Attorney who really just wanted a conviction.
So while I know you’re just doing your job by trying to get me to talk to you I need you to understand my outlook on this which is that my life might depend on me keeping my mouth shut right now. What I will say is that I’m completely innocent of these charges and I think you’re wasting your time talking to me. Beyond that I’m going to keep my mouth shut from this point on and wait for my lawyer. Thanks for understanding.
...As far as any false claims the police might make they don't have the right to do anything without a warrant so I'd use vaguely the same speech as above but substitute in "I've been told you need a warrant to search my person or possessions"
Edit: I just IMed a friend who is a lawyer and he made a good point on searches. When the police claim they have the right to search your person or possessions you want to do two things. First, explicitly say you forbid them to search unless you are legally required to allow them to do it. Second make them take physical action (e.g. if they want to search you tell them you won’t empty your own pockets but you won’t resist if they search your pockets). Do these two things and you’ll never be searched unduly.