It depends on the jurisdictions involved. Many U.S. states allow one-party consent, but several require all parties to agree, so check current law for each location.
The laws of each participant's location can matter, and in practice the stricter rule may apply. Check both states' rules and consider asking for consent openly.
No. Recording consent is separate from publication terms. Agree on naming, off-the-record passages and quotation review, and write the agreement down.
Ask the question at the start of the recording and capture the answer, then store that with a note of the date, locations and law you relied on.
Facility rules usually govern calls and visits with people in custody. Check the facility's policies and applicable law before attempting any recording.