Some states say that if the vehicle was originally equipped with removable doors, then it's legal to drive them that way. Besides Wranglers, that may include the Hummer (H1?) and some step-vans, like some UPS trucks.
Other states (NY?) say no way.
The mirror requirement is usually separate from the doors issue.
Only way to know is look up your state law or ask your local police.
Back in my CJ-5 days, if I took the top off, there was nothing left to latch the doors to.