1) probably fine, if they are good quality AS. Though we did get some flurries in June, it was on the gravel roads.
2) Check google maps, but on the paved roads probably not more than 2-3 hours apart driving time. Once you get to the Dempster there is nothing until Eagle Plains, that's probably the longest stretch. From Eagle Plains you can get gas at Fort McPherson, so not too far.
3)They are good, Tombstone is fantastic. Further up the road there is Engineer Creek which is also nice. Neither were full when we went. Eagle Plains might be worth booking, and they have indoor showers and a restaurant. In Inuvik you probably want to stay in town at Happy Valley, also worth booking ahead. No idea further north as there was no road when we went.
4) The usual, be bear aware. I think you are fine in tents.
5) Tombstone. We did a hike up Angelcomb Peak, incredible views and pretty easy. Everything else you really just take in as you go, stop when you want. If you have more time Camp in Dawson City(the big campground is across the river, you probably want to book it), then drive to Chicken Alaska over the pass, it's pretty cool. Come back down to Kluane. Loads to do there if you are interested in that plan.
5a) car washes! Dempster City has one, and the cheap one in Inuvik is here I think(next to the cheap gas, don't buy at the first one you see, it's, like 30cents more for suckers!):
https://goo.gl/maps/DCaYjowC3coybnpA8
6) I wish I had a mountain bike when I was there. Kayaks? Tougher call, but there are plenty of rivers/lakes.