For 1-3, you may find cocoons, webs and damaged comb on frames. If there none or not many to be seen on a frame, there could still be hundreds of eggs in empty brood cells. The least-affected frames could be reused if you freeze them for at least 24 hours. What are the frames like in your new box? Any drawn comb available or are they brand new, with just foundation or foundationless? The procedure for placing a colony into a different box doesn’t take very long, but will depend on things like how far apart the hives are, the freezing and thawing process if you’ll need to swap some frames back in because you have too many without comb to give them. Always assuming you will proceed deliberately without rushing
Your question about bee health - wax moths fortunately don’t infect bees with any diseases as far as I know. But when their presence becomes an infestation, a bee colony could decide to abscond - which means leave to find a better home. This is similar to a swarm in that the queen will leave with some workers, but gravely different in that they leave brood behind, no workers to raise a new queen, and have to find a safe place to go at this time of year.with little time left before cold weather. They also might not have much honey left to take with them. So, even if your effort to move them isn’t quite smooth or confident, it’s the best thing to do for them!
Re question 5, have all your equipment ready and proceed as efficiently as you can.
Include food for them in the new box, whether it’s honey or 2:1 sugar water, and a small piece of pollen patty.
Close down their entrance to a small gap they can more easily defend against robbers.
Wait about 2 days and check on food, rotate back in any drawn comb frames you salvaged and froze/thawed to replace temporary combless frames. Give more sugar water or honey if it’s gone, monitor the pollen patty to ensure that bees are taking it vs not. Give the bees a chance to have it but if they don’t seem to be taking it, you should remove it so it isn’t an unguarded resource for small hive beetles.
Re Fred’s question, it would be a good idea to put a beetle trap in the new hive too. We can talk about varroa mite management once you have the bees resettled if you need some tips for that