Is it legal to possess a firearm with no serial number if it was stolen the serial number ground off and then recovered and returned by the police?

Answer 1

The correct answer varies from state to state, but assuming the law in your location is the same as the law in mine:

If it's a bolt or lever action rifle, shotgun or a long barreled handgun, yes.

Assault rifle, short barreled rifle, submachine gun, or most any other conventional firearm not mentioned is illegal to own without registration by serial number in most states and countries.

Of course, this can be overwritten with certain licenses, but most of these require the gun to be of a certain age (before they started putting serial numbers on the gun, such as an early Thompson submachine gun) and in many cases must be disabled (usually either the firing pin removed or receiver welded shut.)