I was planning on tilling, leveling, and amending our existing 1000sf backyard bermuda grass lawn this fall. I got a new pup who loves to dig, and decided it would be a waste of time until she grows and I can train her better so I put it off and overseeded rye last week... I need some advice so I can plan again.

I moved into the house a few years ago, and the soil is heavy clay, unlevel and puddles, and really needs some organics amendment.

1.) When should I do this? May is generally transition time that Bermuda will wake and the rye will go bye bye.
2.) When I till, I plan on leaving all the Bermuda rhizomes as much as possible since I need the organic matter. I will have to remove a decent amount of clay soil to add additional organics. Will the Bermuda recover after being torn up or should I reseed? This lawn is real old, so how will the old bermuda mix with a nice strain I seed? Will a newer high quality strain eventually overtake whatever I have? I'd prefer this.
3.) Some more info after reading descriptions of bermuda strains. I have two medium sized dogs running around the back yard all the time, I'm not looking for a putting green. I've also been moving towards the mow-high, water deeply/infrequently, reel mower leave clippings method of lawn. Really trying to go organic but can't until I fix this clay soil.
4.) Suggestions for seed? Any other advice? I know basics about general soil care, not so much about what I'm about to do.

Thanks in advance.