You always have the right to remain silent and to a lawyer whether you are under arrest or not.
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You raise a number of issues which you should discuss with your attorney.
However, there are court decisions that say it is OK for the cops to lie to you the suspect. There are cases dealing with the issue of improper advisement of rights . No cop should lie , under oath, to a court.
The above is not intended as legal advice. The response does not constitute the creation of an attorney client relationship as this forum does not provide for a confidential communication.
If you are being questioned while in custody and ask for a lawyer, the questioning must stop. I see no reason why the law should be any different just because you are not in custody. As a practical matter if you are not in custody and you want a lawyer, what is preventing you from leaving? Nothing. You need a well qualified criminal lawyer. Most police announce the time they are starting the recording and the time they are stopping each time it is started and stopped. If the officer did not follow this practice, this is at least one matter that the defense attorney can cross examine about. There may be many other approaches to cross examining the officer. Effective cross examination will be crucial to showing the officer is not being honest about certain things.