The different batches won't come out identical. In my experience, the first batch is going to have more yeast character as the yeast had to grow into the right colony size. The second batch will probably feature the hops a bit more, not a problem with an APA. It will probably come out a little drier than the first batch as it attenuates a bit further, again not a problem for an APA.
As for the dry-hop issue, I don't think I can offer any advice. Are you removing the hops after the dry hopping? If so, no problem. If you leave them, I would think they would impart a hoppy, and then grassy, flavor to the new batch. If you are dry hopping with leaves/flowers thrown in there, I would personally pitch a new yeast starter with the new batch. If you are dry-hopping with pellets, put them in a hop-bag and give yourself a way to pull the bag when it is time to move on.
I disagree with BrewStooge on the wiping the kreusen ring issue. That stuff may look like nasty sludge, but it is sterile. When I pitch onto a cake, I disturb the inside of the bucket/bottle as little as possible. Why risk introducing new contaminants?