E Sinister Spiderman is a wildcard transporter. If he copies batman enemy and carries Ra's al Ghul, does he get an attack bonus when making a transporter attack?

What if the same thing, but replacing transport attack and spiderman with emeral empress and running shot?

1) no in a transporter attack, and HSS attacks, figures being carried are considered off of the map until the entire action is resolved. So, the carried character will not be available to share AV's with the transporter (or HSS character).

2) Yes, because when the movement part of the running shot (or charge) is completed the passenger is placed back on the map before the attack and is available to share their AV.

What if Spiderman ends his movement with his attack? Or if he makes his transporter attack when next to Ra's, but not carrying him?

The attack is still during movement, even if it is the very last part of movement, so any carried figure is still off the board.
Then he can use the BE TA.

Here:

When using charge or running shot: yes, because the move is completed before the attack.

When using Transporter Attack or HSS: No, because the attack is made during the move action, even if the character will end his/her movement in the square from which he/she is making the attack.

As for your second question: Replace, then modify.

Spider-Man moves up next to Ra's, replaces his AV with Ra's, then modifies the AV by -2 per the rules of the move+attack ability.

On his first click, Spidey would then replace his 9 AV with Ra's 11 AV, then subtract 2 = 9.

Thanks guys!