My biggest concern is that you may be trying to do too much, and you will be spending a lot of time/money in transit and checking in/out of hotels. Plus you seem to have a fairly large group of travelers, and logistics can get complicated.
I'm planning a 10 day trip as well for June. Seoul is large enough that I ended up extending the stay from 4 nights to a full week. One night in Busan is perfectly reasonable. For Jeju, if you are spending that much money on flights to get to Jeju, you definitely want to spend at least two nights there to make it worthwhile.