No. The Red Cross has no more access to military prisons as they do to state and local prisons. The military will inform families of their serviceman's condition if they are not able to do so themselves, when they are killed, wounded, or injured for example. Anything else is the responsibility of the serviceman.
Not as I remember it.
I once guarded he brig on ship for 30 days, and while there was no Red Cross on board, the families of the 2 guys held were notified through the Red Cross of them being in the brig
Either the Chaplain told them or our C.O. told them, but it was the Red Cross that told the families.