You may not realize it, but Catholic elementary and high schools are fully funded and part of the public school system in Ontario. They are managed somewhat separately, with their own school boards and trustees, but they are subject to all of the government oversight of a public school system. Also, because they are supported by taxpayers, enrollment may be open to non-Catholics. The local Catholic high school probably has as many non-Catholics as Catholics attending. The non-Catholics agree to attend religious classes.
The publicly funded school system is set up as follows:
- Public School - non-religious elementary and high school education, open to any student within the district boundaries served by the school
- Public French Immersion Schools - for students who wish to learn in French but are not necessarily from French-speaking families. Often housed in the same building as the public school program, sharing facilities and teachers between the 2 programs
- Public French Language - for students from families with at least one parent who speaks French
- Catholic School - separate school board but publicly funded elementary and high schools. Non-Catholic students may be enrolled.
- Catholic French Immersion
- Catholic French Language
French Immersion is often a very good option, if you are interested in it. You don't have to be fluent in French for your children to attend. Most parents are not. Most programs start in Kindergarten or Grade 1 (not "1st Grade" 
), known as Early French Immersion ("EFI"). Some school boards offer middle immersion starting in Grade 4 and late immersion in Grade 7. so it may be an option even though your children are older than kindergarten age.
I don't think you will have trouble finding gymnastics programs. They are pretty popular at various levels from recreational to competitive.
Many years ago, the University of Western Ontario in London had an excellent, competitive cheerleading team. They placed well at competitions in the U.S. I don't know if that's still the case. If you wanted to research competitive Canadian cheer teams at the junior school level, it might be a place to start. If the UWO team is still competitive, you could see where they are recruiting for their team members.
Vaughan is a comfortable commuter town. I don't think I would describe it as rural. For that you would have to go further north or east or west. I could be wrong though, since I've only driven through it. From the car, it seems to be pretty urban/surburban. The houses are larger than many in the core area but I don't think they are much bigger than houses in North York or Etobicoke etc, that is the suburban areas of Toronto proper. I think Vaughan had quite a bit of political upheaval in the past couple of years but that probably doesn't concern you. On a day-to-day basis, probably the biggest drawback is the drive into Toronto if you have to commute anywhere downtown. I looked up Stuart Dean and found an address for 1 Yonge St. If that's his office, then he is looking at a long commute every day from Vaughan all the way down to Lake Ontario.
In case you are interested, the older, original Little Italy is located along College St. in downtown Toronto and is still a thriving area, and there is also the Corso Italia near St. Clair and Dufferin.
One of Toronto's most attractive qualities, IMO, is that it continues to have thriving residential communities in the inner city core as well as the outer suburbs. Depending on your price level, you may find a 4 bedroom home with a fenced-in yard much closer than Vaughan.
). It's 30 mins-1 hour by GO Train (our commuter trains) to downtown Toronto or you can catch the subway from Yorkdale Mall to downtown. I won't lie, Toronto is not exactly famous for great public transit... it works, but it won't save you loads of time and money like it would in other cities. 
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