BLM land is usually signed as no off road access, if that is the case. The Forest Service rule for both camping and activities such as firewood cutting is within 300' of an official road. Do use existing sites when available, but don't be scared off of creating a new one if appropriate. I've followed that same distance on BLM land just as good practice.
There are lots of pre-existing campsites that are more than 300' from an official road, it is shame that Coconino is doing that in Cottonwood.
The 14 day rule is also pretty universal though the days per month and minimum distance to move vary widely. There is an exception to the 14 day rule for folks working on the forest, like herders, firewood cutters, etc. I suspect a gathering permit, garnered from the local office for a nominal fee, would suffice.