New lawn in sections?

My lawn was heavily damaged by a Bobcat (the mechanical kind). I have big patches of dead grass and areas where the ruts are 12 inches deep. I bought a bunch of loam and spread it out to take care of the ruts. For the areas that were damaged but still had some grass, I roto-tilled them. I have two questions:
1) Can i seed a new lawn in sections, doing one half now and one half in the fall? (i'm in new england) I know it's late for the spring seeding already, the weather has been unusually cool. .
2) After roto-tilling the grass sections, I have thousands of small clumps of grass that are impossible to rake up. can I loam over them and then plant or do I need to remove all the tilled-up old grass?
Any advice is greatly appreciated.

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