If a ship looses enough movement to get a -2 modifier while beating does that mean it cannot move at all in subsequent turns (unless the wind direction changes)?
On its next activation, it can roll on the sternway table (see 3.6.9) which allows it to rotate to a new heading and get underway once again.
How about tacking? Can a ship with no movement tack and hope for a lucky roll where it gets enough movement points to make it into a reaching position?
No, a ship that has no MPs owing to rigging damage can only roll on the sternway table.
Can you tack only at the beginning of a ships movement or can you tack at anytime during the movement?
Only at the beginning of its movement. See the next to last sentence in 3.6.6.