Chose Army Infantry Airborne. Will I be assigned to and airborne unit?

If this is an active duty contract, and the contract says 82nd (or some other airborne unit), then yes. If it does not specify a unit, then it's needs of the Army, you could end up anywhere, even with your Airborne training. 300 to 500 Soldiers graduate Airborne every week, not all are from or going to an Airborne unit.
But you can volunteer for RASP at Basic or Airborne School.
If this is a Guard or Reserve contract, your para/line in a unit is already determined.

Simply- Yes.

Probably, almost certainly, but not 100%. Airborne who are initial entry will sometimes end up in non airborne units.
Your best bet is to volunteer for RASP after Airborne and become a man among men -
A Goddam Airborne Ranger! (you better ask somebody).