1) If a player activates the FTL control during the turn and jumps, then at the end of the turn they draw a crisis card. If said card has a "jump advances" then does the jump advance from the start? Not sure if we're doing this right.
Yes, the jump track will have reset before the Crisis phase of the turn. Same for a cylon attack -- the basestar will show up on a fresh jump clock!
2) If a cylon draws a crisis via the Caprica action that is a "current player chooses" does the cylon choose? If the Cylon draws one that is "president/admiral chooses" and the prez/adm. has the choice of discarding some cards + making the current player discard, does the cylon discard?
Yes, the Cylon is the "Current Player" for all purposes of a crisis card. It's a pretty messy case, and one thing the game does not really handle well at all, in my opinion. The only major exception is that the cylon player cannot be sent to the brig/sickbay.