I'll answer the best I can.

Pup - Depends on the dog. I have seen plenty of dogs in canoes. I would never bring mine, though - I don't think she would sit still in the canoe. Then again, my kids have trouble with that, and I still bring them...

Gear list - this thread should give you a good idea of stuff to bring - https://www.paddleplanner.com/forum/default.aspx?g=posts&t=6603. Packing for a canoe trip is a lot more flexible than a backpacking trip. You can bring more and/or heavier things because most of the time is spent in the canoe, and it is easier to paddle than walk.

Bears - I've been going for over 25 years, and I have never seen a bear. They are a lot more secretive than most people think. There are "problem" bears once in a while, and that most often occurs where there are a lot of people (the bears have learned that people = food). Take care of your food properly, and bears most likely won't be a problem. 

Route - Easy routes usually equate to more paddling and less portaging. Routes like that abound in the Quetico. To offer specific suggestions, we would want to know a little more, like whether you want to travel every day or take a day off, whether fishing is important, or whether you want a specific type of campsite (e.g. sandy beach?). (FYI - Campsites are first-come, first-served, so that can never be guaranteed.) I will offer some general suggestions for entry points. Also note that most of these are large lakes, and wind can be a factor.

• Pickerel Lake - It is 20 miles (32 km) from the put-in at French Lake to the western end of Pickerel, no portaging required. Plenty of variety and places to explore within that 20 miles. You can go as far (or as short) as you want. Fishing, sandy beaches, hiking trails, scenery - pretty much has it all. Just have to watch out for the wind since the lake can really get rolling.
• Cirrus Lake - Three portages (medium length, but well-maintained) will get you to Cirrus Lake. From there you can go over 14 miles (23 km) without another portage.
• Quetico Lake - Two portages (one medium, one short) will get you to Quetico Lake. It is 13.5 miles (22 km) to the eastern end of the lake, but you also have the option of going north. If you want a loop, you can go north on Quetico Lake, take one of the portages to Cirrus, and then head back towards the entry point. Or vice versa.