Ages 7 and 9 are very, very young for this type of activity, and therefore I would think that any climbing club would have extremely limited expectations as to what they can/should do. Personally, for kids this small, I would be happy if they could climb on top-rope, not freak out when they're too high off the ground, and, when it's time for them to be lowered, they should have the psychological willingness to lean back and weight the rope. If a kid age 7 can tie in with her own rewoven figure eight, properly dressed and with an appropriate tail, then that's awesome, but she should be checked every time by someone who is competent. It's fine if a kid that small needs someone else to tie them in.
They should know the standard voice commands used in your country, and they should be used to wearing a helmet, including at times when they're not actually climbing but could be hit by rockfall, a dropping phone, etc. They should understand some basic climbing etiquette, such as not to step on ropes and not to walk under a top-rope setup.
For their own protection, I would teach them that when they're toproping, they should do two things before climbing: (1) check that there is a knot in the other end of the rope (so that they can't get dropped if the rope is too short), and (2) make sure their belayer understands that they expect to be lowered off (that they will not be rapping off). They should also be used to checking your knots and harness buckles, and having you check theirs, before every climb. They should understand how to check whether there is slack in the rope, and what voice command to use to alert the belayer when there is too much slack ("up rope" in the US).
Teaching them to rappel is IMO a bad idea at this stage. Rappelling is inherently dangerous. If you read Accidents in North American Climbing, every year a huge percentage of the accidents are rappelling accidents. There is no need for them to know how to rappel when they have such a small amount of experience. Rappelling would be something they would need to do much later, in a situation where they're leading and need to clean. The second-hand backup you're teaching them sounds pretty standard, but I think it's just extremely premature to be teaching it to them.
Habits, once ingrained, become hard to break. Therefore I would not teach them to do something weird and idiosyncratic when they get to the top of a climb. Whatever procedure you're having them follow to set up to rap off, it doesn't sound like any procedure that I would want to have them do in any real climbing situation.
I assume that teaching them to ascend the rope on a Prusik is simply an expedient you're using because there is nothing for them to actually climb. This is actually a good skill to have for the future, but it's very unusual for kids this small to learn it. Consider attaching some kind of holds to the tree so that they can actually climb.
Nothing you say about your anchors immediately raises red flags for me, but it's hard to tell over the internet. You don't say whether the slings are cord or webbing, or what kind of knots you're using. I don't know how many times you're going to climb on this setup, but if the number of climbs gets very large, you want to make sure the permanent carabiners are not wearing out from rope friction. You want to inspect them frequently to make sure the screws are still locked and the biners are not cross-loaded. I would avoid leaving the climbing ropes continuously outside and exposed to ultraviolet. Replace the slings frequently, especially if they start to look stiff or faded.
My personal rule of thumb for using a tree as an anchor is "five and alive," i.e., the trunk (or branch in your case) should be at least 5 inches in diameter, and it should be healthy, with green leaves growing from it. The attachment should be as close to the roots as possible, or in your case as close to the trunk as possible, which does seem to be the case. If it meets these criteria, I'm OK with using it as an anchor without any redundancy. However, if it was a setup like this that I was using with small kids, over and over, I would try to set up some better redundancy, so that even if the branch breaks, there is something else that your anchor is attached to.
In general, it sounds like you've gotten your knowledge from reading rather than from people who actually climb. Lots of people have had to learn to climb this way. However, you would do really well to seek out more experienced people to mentor you and your daughters.
Guides often belay two clients at once, but it's less common to see other people doing this. It's more complicated to keep track of everything properly, and when you're just toproping on single-pitch climbs, there's normally no need to do it, because everybody just takes turns. Since you're not very experienced, I would just avoid doing it. Make sure you have the basics of belaying one person figured out correctly, so that when you do it, what's getting programmed into your muscle memory is correct.