2) I couldn't help clingsticking this sticker from the banana onto my forehead.

3) FThe father can't help but try to set his son set free from incarceration.

6) Carl couldn't help doing a takeoffn impression of Scooby-Doo.

2) I couldn't help clingputting this sticker from the banana onto my forehead.

The best verb would be "stick" here, but often in English we try to avoid repeating similar words next to each other. "Stick this sticker" sounds awkward, though it's not technically wrong and some people may still say it.

3) FThe father can't help set his son set free from incarceratiprison.

Incarceration is an official word that is not often used, and it sounds awkward here.

6) Carl couldn't help doing a takeoffn impression of Scooby-Doo.

I think "impression" is what you mean here. Is Carl imitating Scooby-Doo? If so, "impression" is correct.

2) I couldn't help clingsticking this sticker from the banana onto my forehead.

"Cling" generally only takes direct objects, like "the sticker clung to the wall."

3) Father can't help his son set free frombut try to help his son avoid incarceration.

This is unclear. Based on the premise of the exercise, it sounds like you mean that the father had to help his son avoid/escape prison, but it reads as if the father was unable to help his son. For this type of structure, you need either the gerund (-ing form) or the infinitive of the verb prefixed with "to."

4) Ben can't help being disguised as a staff member.

This is technically correct, but absent other context sounds strange.

6) Carl couldn't help doing a take-off of Scooby-Doo.