Situation...

3rd and 5 from A's 40.
A1 sweeps around the side on an option. He runs past the line of scrimmage to A's 45, where he tries to pitch to A2...
BUT the pitch is forward, and RB-A2 drops it. The ball hits the ground, A2 picks it up on the hop, and continues to run downfield where he is tackled at B's 45 yard line.

Illegal forward pass. Dead and incomplete when it hits the ground. A is penalized 5 yds and loss of down from the spot of the pass.

You have an illegal pass and an incomplete pass.

What if B recovers. Is it still an incomplete pass or is it a fumble?

B just recovered a dead ball: the incompleted pass. Still A's ball after penalty enforcement.

Why would you think it was a fumble? The QB's intent is immaterial - he threw a forward pass.

Rule 3-4-4f
The clock shall be stopped when...a legal or illegal forward pass is incomplete.

Incomplete illegal fwd pass.
If B accepts: 5 yds from spot of foul and LOD, 4th & 5 from the A-40
If B declines: The ball is put in play at the spot of the illegal pass (they would never decline this)

Expanding on the situation, let's say this is a penalty that does not include LOD.....so accepting would leave 3rd & 5....and declining would leave 4th & inches or 1st & 10.

Could B request measurement before deciding if they will Accept or Decline the penalty? If the result of the play would leave a 4th & inches situation, B might want to decline the penalty and if it would result in a 1st & 10 they would obviously accept to make it 3rd & 5.

Is it supported by rule that B may request measurement before deciding on penalty options?

Fodder for a completely different thread... but of course the team deciding whether to accept or decline a penalty is entitled to all of the relevant information.

Canadian Ruling
 

Offside pass (subject to penalty). And a live ball.

B will require A to scrimmage from the A-45, with a first down due to yards gained.

Same as for a legal forward pass - he needs to posess it before it touches the ground.