Find a class III dealer first and let them guide you through the process. That is what you will be paying them for, whether you use the service or not. Although the process is pretty generic, every local has its ways of doing things. To purchase a class III firearm prior to having all your ducks lined up in a row can be a felony.
If you move to a non Class III state, the firearm must be sold or surrendered (outside the new state of residence). ATF must be notified about ANY residence changes(state to state, not within a state of residence), but if I'm not mistaken, LE permissions (signatures) are not needed and no additional charges (stamps).
Signature is not required for corporate ownership, but you do surrender your 4th Amendment rights when you do so. Corporations have no Fourth to stand on. AFT can come and inspect premises at will.
And yes, most ranges are no full automatic firing allowed.
Here, found this cool link that has most all the information that you need.