It doesn't say to treat it like a mech. It says "The Bomb acts sort of like a mech". It says it can be pushed or towed, and that it can squash minions while taking damage, but it doesn't say it can be repaired.
So, I would say no, you can't repair the bomb with auto wrench, as that would just make it too easy as you could just push the bomb through a wave of minions, taking one damage and then repairing.
This is correct.
Auto-wrench will not repair the bomb.
Schematics turn on when you prefer as long as the gear requirement has been met. Some schematics say specifically on the card that they can only be used at a certain time ("play before you execute your command line") but other than that, you can execute your schematics as soon as they are unlocked.