It should be a 3 minute spin up from liftoff. I haven't timed it carefully but I haven't noticed an obvious variation. If you do find one please post a track and I'll submit a report.
Between Maverick shots, I haven't found it necessary to reset to T0, only to cycle the safety, then do T1 and TV to lock. Can you check this again?
I agree with you on the "direct TV fix", and I plan to submit a request for that capability. I don't think it needs to be a special keybind, though. We just need the Mav to assume an implicit T1 any time it goes from T0 to TV.