Sorry to hear that, but well done on getting the survey, another member here posted not so long ago that her son's boat which he had had only a few months and not got surveyed, started taking in water and the short version is, the hull was a state and will cost a fortune to fix, so you've dodged a bullet there.
I moved onto my boat three days ago with my cats, they have settled about the same as they have on house moves so far, no worse for the rocking about and maybe even a bit better as it's been a smaller space for them to get used to...
I am moving to a different, more cat friendly access pontoon on Thursday when they will be allowed out for the first time since we got on, but they are all keen to go out already and two of them have already had a little look around on the stern and one walked all round the boat on the gunwhales (unplanned, and much to my horror!) and came back in to consider her findings and still seems ok!
Advice I have had and am going to utilise- get a couple of extra rope fenders and hang them so that they are actually partially in the water, so the cats have something to haul themselves out on if they cannot get out on a bank.
And keep a big net on the roof to fish them out with for any impromptu swandives!