I know this is an oldish thread now, but another Beaver leader here!
As others have said, it is not at all uncommon for Beavers to still be in pull ups. We make sure that they have privacy to change into them without others being aware, and are able to dispose of them next morning. It may be fairly common, but we make sure that we deal with it sensitively, and the right to privacy should be a given at any nights away experience. We also make sure that we are prepared to handle any accidents.
Homesickness is another worry that parents sometimes discuss with us. We do find that the Beavers are actually a lot more resilient than expected, and we have a range of tactics to deal with this. Particularly, giving a homesick child an appropriate (and fun) job to do is often very effective. We also make sure that they are well fed, warm and as dry as possible, as well as getting them to bed at just the right time and encouraging them to have a lie in, as keeping them comfortable and occupied helps to keep them happy. Of course, a child could be picked up or phone their parents if they absolutely need to. We have found that phoning parents can actually make things worse though. - and if a Beaver is picked up it can cause the others (who were previously happy) to become homesick themselves. So, it tends to be a last resort. Having said that, if it is the right course of action (especially if a child may be ill) I will not hesitate to do the right thing by them.
Sleepovers and camps are of course not compulsory, but you may find that some leaders ask them attend at least one in order to attain their Chief Scouts Bronze Award before they move up to Cubs, as it is one of the major requirements.
We will soon be taking around half of our Beavers (some of whom will have only just turned 6) away for 2 nights along with our Cubs and Scouts at our County water sports center, and I hope to get them camping in tents for a night later in the summer too. I have to admit that I am really looking forward to it, although I am glad it is on May bank holiday weekend, as I will need a good 24 hours to recover myself!