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.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
Add image file
Upload is a website for homeowners and building and maintenance pros. It is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.