Can first time convicted felon ever get his gun rights back in the state of tennessee? Can i use a cross or compound bow?

Only convictions or adjudications are counted against you to create a federal firearms disability. If you are offered diversion, the usual procedure is to put you on probation while you complete the agreement. If you complete the terms, the charges are dropped. Dropped charges are not a conviction or adjudication (at least where I'm from). If you are adjudicated or convicted or both, state felonies and major misdemeanors can be addressed through executive clemency. Federal felonies and major misdemeanors require a presidential pardon which is statistically very rare.