"Since my cabinets go all the way to the ceiling, do I have to basically do everything since I need the ceiling and floor done before I can get an accurate height?"

No. Remember, your cabinets have to be installed level and plumb, no matter what your floors and ceiling are doing. I'd check each with a water level or laser and if you're out excessively on either, I'd calculate what the dimensions will be after the fix and finishes. Leave a little play.

You should be able to rig up your old cabinets, sink, and countertops into the dining room without too much trouble. It'll be worth it.

Thanks Joseph. What do I do with the "little play" when everything is said and done? Are there any solutions for gaps between the cabinets and the ceiling apart from crown molding?

"What do I do with the "little play" when everything is said and done? Are there any solutions for gaps between the cabinets and the ceiling apart from crown molding?"

Using a spacer and setting it back in from the edge of cabinetry is the way most will do it. This spacer could be 5/8", 3/4" or whatever. Figure out how much your ceiling deviates from one side to the other. If it is more than a 1/4" you will probably notice it when using a thinner spacer. Using a wider spacer will allow you to scribe the spacer so that the edge that meets the ceiling follows the deviation and the straight edge meets the cabinetry.

Also, don't assume that "7 weeks lead time" is a sure thing. Even if it turns out to be correct, there might be some problem(s) with the cabinets that require re-work and more time. Better to order sooner rather than later, in my opinion.

"What do I do with the "little play" when everything is said and done? Are there any solutions for gaps between the cabinets and the ceiling apart from crown molding?"

I was hired to help a guy install some frameless cabinets that the customer wanted mounted all the way to the ceiling. The problem was, the ceiling was way crooked. You can have your cabinets touch the ceiling at its lowest point and fill the ever-widening gap with some sort of tapered trim or you can have the ceiling leveled before you start, but you cannot have cabinets installed flush to a seriously out-of-level ceiling.

Oh, and they must hang plumb too, or the cabinet door may strike the ceiling as its opened. You never forget the first time this happens, believe me. A customer will probably never notice a line of caulk taking up a "little play", but she'll notice the door hitting the ceiling undoubtably.