If your UV-82 is locked, which it sounds like it is, it will not transmit on GMRS frequencies (or any non-H.A.M. frequencies), and yes, it can still receive and listen to those frequenices. If your UV-82 is not locked, then it will also transmit on GMRS and other non-H.A.M. frequencies.
The DB20G is a mobile GMRS radio so of the box it will transmit on all of the standard GMRS channels EXCEPT for channels 8-14, so because channel 7 is not between 8 and 14, it should also transmit on channel 7. If it does not, then you are either doing it wrong, you screwed up the programming, or its defective - and i doubt it is defective.
ANY hand-held GMRS radio will talk back and forth with FRS radios. Any mobile GMRS radio will talk back and forth with FRS radios on channels 1-7 and channels 15-22 .