For what it's worth, here is my perspective....I'm pretty pragmatic when it comes to this kind of thing. I don't believe children at that age should be allowed to just drop out of an activity that they chose on a whim. It is not like you are trying to get her to go do something that may not be age appropriate. Your children are old enough to make a choice like this and comprehend what it means when you sign up. It is now a responsibility to attend. I feel that as parents it is our role to teach our kids to value their commitments and respect others. Therefore, once they requested to sigh up, and you have done us, I wouldn't even ask them again whether they still want to go. You are opening up the opportunity for them to say no. I would instead just announce that this week on such and such day is your art class. How fun! And then take him/her as planned. Next time when you are signing up for a class, I would consider the following:
1. When you ask if s/he is interested in signing up, make sure you clearly state that signing up means that they are committed to go to the class (barring illness,etc, of course which you do not tell them in advance). So, they need to pick carefully.
2. If they put their foot down and really refuse to go to the class after having signed up. Rather than force them, give them a choice. "Well, the class cost me $30, so either you go or you pay me back the $30 from your allowance/piggy bank, etc." If they don't have $$ (e.g., gifts from grandparents in their piggy bank or similar), then tell them they need to find something to give up in exchange - such as a toy. The object is for them to understand they have a responsibility once they make a choice.
3. The next time they ask to sign up for an activity, tell them you really need to think about it because the last time they did not want to go and that was wasteful. If they had ended up going to the class even if reluctantly at first, then reset the expectation and sign them up. If the time for class comes and they again say they don't want to go, well, that's the end of that. Collect for the class and then next time they want to sign up, they don't get to. Period. Immediate consequence. Let them earn back the opportunity.
When it comes to the whole soccer situation. I would not sign him up again, even if he asks, unless he played out the season joyfully and really is interested. Make him wait it out another season and if he asks again the following year, set expectations and sign him up if he understands and is onboard with the expectation that he will not change his mind or drop without giving it his best effort.
By the way, I read someone say it is not about the money. Well, it certainly is not all about the money, it is about responsibility, but kids this age also need to learn the value of money. So, in a way it is also about the money and teaching them to be responsible to themselves, to others and in making small financial decisions.
Hope that helps!