I've mountain-biked the Apache Trail from Canyon Lake (Tortilla Flat) to Roosevelt Lake. If the road is open generally, I'm not aware of anything that prohibits biking it. It is a tough haul, with a surprising amount of elevation gain. And, of course, this time of year, it will be roasting. I parked near the store/cantina at Tortilla Flat, but mine was just a day trip--not overnight. Also, parking there starts you off with 1100 ft. climb in the first 5.5 miles up to the "end of pavement"--where the road bikers who head out to Canyon Lake generally turn around.
There is a parking area and bathroom just before you descend to Fish Creek Hill. I assume you could park there with a Tonto Pass(?). I'd probably recommend that as a better starting point. I don't know how "safe" it is there in terms of car vandalism, but haven't heard any horror stories.
As for safety of bike riding the route, the road into Canyon Lake itself is much more dangerous, in my opinion. I did that too once on my road bike. It's a cool ride, but there are no shoulders and tons of blind curves.
The dirt road involves much less risk of getting hit by a car, IMO. Just exercise caution and be a defensive bicyclist--and make sure your bike is in good condition (esp. brakes on Fish Creek Hill!). Be prepared for a dusty, wash-boardy conditions, going up and down/in and out of innumerable draws.