I'm sure this has been asked before, but I must know: could I have one ship with TS and lock a ship and all of my OS-1s get to shoot that ship during a weapons disabled period?

No. Os-1 is very specific in requiring that you have the lock on the enemy ship. Targeting Sync does not allow you to treat its ship's lock as your own, so it does not interact with Os-1 at all (outside of maybe allowing your Boats to keep their own locks, I guess).

the new faq entry on atc also suggests this-

A TIE Advanced equipped with Adv. Targeting Computer

needs to have a target lock on the defender in order to

use the ability. The TIE Advanced cannot use a target

lock on the defender from a friendly ship equipped with

Targeting Synchronizer for this requirement nor can

it spend that target lock if it uses the ability of Adv.

Targeting Computer.

What about Omega Ace? TS says: If a game effect instructs you to spend a TL...

And you spend the lock for OA...

Yes it works for OA. Who only requires the spending of a lock...

1. You can spend the lock as if it was your lock - so it works for omega ace, concussion missiles, things like that.
2. You cannot 'count the lock as yours' for stuff that requires you to have the lock but not spend it (other than the ATTACK: Target Lock header because that's specifically called out) - so it doesn't work for OS-1 arsenal loadout, Advanced Targeting Computer, Omega Leader.

Compare the wording on Targeting Synchroniser with the wording on Shara Bey's pilot ability. There is a reason she is a Rebel pilot and all the abilities listed in (2) above are on imperial-only cards.