That was my thought when I read the first page. IANAL, but, as I understand:
You can of course say absolutely nothing when arrested. You can stay mute when the police approach you on the street. There is no law requiring you to talk. (Except, when there is reason to believe you might be involved in what might be a crime, you may be required to identify yourself. IIRC otherwise, failing to identify yourself is obstruction.).
And, once you are arrested, if you don’t say “I want a lawyer” the police can keep trying to get you to talk and keep asking questions. Even if you say “I choose not to talk to you any further” they can keep trying.
Once you are under arrest and ask for a lawyer, then the police must funnel all further questions through your lawyer. Even then you (or he/she) can refuse to provide answers. It’s only when you get to the witness box (as a defendant, your choice) you must say “I invoke the fifth”.
There’s the whole other part. You can’t get on the stand, say “I didn’t do it” and plead the fifth beyond that. Once you open up to testifying or answering questions about something, you cannot clam up on the topic half way though.
If you lie to police, and it is intended to misdirect the investigation, that is obstruction. Telling them that you had pizza last week not burgers is not a crime, unless it is relevant to the investigation. Presumably telling them you didn’t sleep with the boss’s wife is not a matter of obstruction unless it’s relevant. With the FBI, YMMV.
The interesting thing about immunity is - in the USA there is state and federal law. So immunity under one does not necessarily shield a defendant from the other jurisdiction. Interestingly, it has no bearing on civil suits; AFAIK once immune, you can’t use the fifth to evade questions in a civil suit, but I assume you can refuse to answer - unlike criminal court, you don’t have to answer anything, but that refusal (or lack of a helpful answer) can be held against you.
So for the case of Epstein in Florida with his first charges, that the feds charged him I understand did not stop the state from also doing so. They just chose not to. IIRC there was some question about double jeopardy.
A reverse tactic, NY State is suing the Trump organization in civil court, and there is concern that the upper management can either answer and their answers could later be used against them if there are criminal charges - or refuse to answer pleading the fifth, and that fact can be held against the company in the civil suit - suggesting higher penalties may apply.