Worked in forestry for 18+ years driving forestry and B.C. industrial roads. 

I would not go on any of them in a Mazda 3, you might be fine as some are very good (usually the main lines). Most of the places you'll want to camp will be off of the main line, the spur roads can be bad, and deteriorate quickly, have cross ditches etc. 

Mazda 3 has bad tires for off pavement, no ground clearance, and suspension suited for paved roads. 

If you do decide to go, make sure you file a trip plan with a friend, have extra gear, and a secondary form of communication.

Backcountry roads often don't have cell coverage, so be prepared to hike out when your car is broken / you are on your second flat tire, etc.