Will going Airborne lead to automatic PCS?

You will go to Airborne School TDY enroute to your next unit, which will be airborne.
Airborne school no longer accepts school seats from non airborne units, and any time the big army pays for training you'll PCS afterwards.

Either could happen. Often times Soldiers in that situation have been afforded AB trainjng as some sort of recognition which usually results in a normal DEROS to an AB unit. Without more info that is the best I can do. Thank you for your service.

In most cases, depends on your location, amount of time at current duty station and when you are projected to rotate. If you are a single soldier, you are only required to do one year on station unless you were locked-in upon arrival for two years. Call your branch manager and ask him or her. They can tell you when you are projected to PCS. Then it doesn’t matter when the school date is... Right? Could always request to attend as a School en route as long as you pre-coordinated everything with your branch and CSM...