We were there on a Friday in mid June of 2019.
1) There are no toilets as you mention. The boats do have a toilet, but the boat is not there while you're ashore. They make a quick approach to the quay, offload, and leave, to make room for another operator. You're only choice at that point is behind a rock. There are resident docents/rangers that might arrange a trip to their facility but I have no idea if that's a possibility. You might hear from someone else...
2) In 2019, you could detect the coming arrival of Star Wars fans, as many shops were bringing out their life-size cardboard photos of Han Solo and storm troopers for photos, but the craze was in remission in June. There are 15 boats, each with 12 people, so the 180 folks are staggered over the entire day. The climb up the stairs to the beehives is tiring, so it can get a little crowded as you try to pass people that are slow (or fast people pass you). The trip down is much faster except there are people headed both directions and they sometimes knot up.
3) We didn't even know there was a museum. I guess if your trip is cancelled by bad weather, the museum would be an alternative. There are docents on the island to answer questions.
4) The best part of the trip is visiting with the puffins. You see lots of varied species from the boat, but if you sit on a rock, the puffins come right over and start interacting. They stay a foot or so away, but they are SO CUTE. My wife did a cursory visit to the beehives so she could spend most of the trip with the puffins. She set her scrunchie down for a moment to straighten her hair and a puffin ran up and stole it. Of course they are only there in the spring but they should be there in May.
Skellig Michael was the #1 highlight of that trip. Without the puffins, difficult to rank.