Yes, Battleaxe Rd is the start to the eastern entrance of the Coke Ovens via Superior. I don't recommend it as a route - too long. I would though recommend walnut canyon for hiking. You can drive it as long as you can descend a 45 degree drop that is about six feet deep.
You do need a permit to obtain the combo for the gate. The Mesa Ranger Station issues them free of charge.
I may be taking the Salt River Canyon again next weekend. Will let everyone know as soon as I get something put together.