The first question I have, is have you or possibly the people in the house before you use rock salt or deicer on that driveway? If so, you won't want to put your garden there.
I would be leary of using compost that have flecks of garbage in it. You don't know what that garbage is or what sort of chemicals may be in it. I would assume it would be ok for a
flower bed, but you are going to be eating those veggies.
Try starting your own compost. It isn't as hard as it may seem.
I would start looking at your current soil. Maybe add some manure to it after your get it turned and work that in.
Alot of really serious gardeners will tell you to get a soil sample and send it off for testing.
As to how deep you want your bed, that would depend on what you are planting and how. Most
seeds only need sown 1" deep or less. If you use plants instead of
seeds then you need to be able to dig a hole for them. Your best bet I would say would be around 6", but I never really paid attention to how deep my dirt was. My brother-in-law will be plowing mine. My dad always used a rotertiller(bad spelling)So you need enough loose soil to plant in and some extra for roots. 6 inches should be plenty.
No matter how much you turn that soil you will have bugs and
weeds. It is a fact of nature, no garden will be without its bugs and
weeds.