Using E-Transporter Spider-Man from Sinister and assigning Feat Cards, Lunge & Shake Off.

Will the automatic breakaway granted by Lunge activate the damage delt to opposing characters delt by Shake Off???

If not, then why?

No.

Because of this from the player's guide:

A character that automatically breaks away deals no damage with the Shake Off feat (this includes when using Phasing/Teleport). A break away roll is required to use Shake Off.