Does 'Acquiring a lock' count for the r3 astromechs ability?

If I have Jag with R3 and someone shoots Sinker right next to him, does Jag get 1 lock or 2 locks?

Furthermore, if I use Sync Concole to spend one of those locks and let another ship Acquire a lock, if that ship has r3, does it trigger?

No. Acquiring a lock is not performing a lock action.

R3 Astro says very specifically, "after you perform a [Lock] action, you may aquire a lock." In the Rules Reference (1.0.2) on page 12 under Lock you will find this line:

• If an ability instructs a ship to acquire a lock, this is different than performing a [Lock] action. A ship that acquires a lock without performing the action can still perform the [Lock] action this round.

Images of the cards in question for reference:

It's perhaps worth noting that whilst you don't get to acquire two locks at the same time with anything other than a lock action, acquiring a second lock will not break the first, regardless of how you acquire the second. The first clause of R3 is independent of the second.