$500k will get you pretty much anything you want in Kingston, short of a waterfront mansion. Real estate in this city is cheap.
Kingston is served by the French public school system (CÉPEO)--a high school & elementary school. http://www.cepeo.on.ca
Based on all you said I'd recommend the McBurney Park area, because:
-Just north of lower downtown, where all the trendy shops are
-You can walk
to Queen's. About a 15-25 minute walk depending on where in the area you are. But if you want to bus that's an option too, the area's main bus routes (#1 & #2) both go to Queen's.
-You're about 2-3km from the base. Easy bike ride.
-Queen's is nearby so students will certainly rent up. The going rate for student rentals in McBurney Park is about $400-$500 per bedroom per month.
The expense of housing in the area depends on age. It's an old neighbourhood, goes back to the early 19th century, and some of the houses are in need of renovations. Depending on quality of housing you're looking at anything from $150k to $350k.
Another option would be Sydenham neighbourhood (even closer to Queen's, about the same distance to downtown--literally on the other side of DT from McBurney Park) but there you'll be pushing the upper limit of your budget (it's a ritzy area), and if you're by the water you'll actually exceed it. Also student noise is pretty bad down there, if that's something that bugs you.
EDIT: Here's a map.