Yes all MTA offices will reopen July 6, 2020 according to the employee text that was sent out. As far as DMV in nyc, read https://dmv.ny.gov/more-info/dmv-reopening-guidance.
I know for a fact bus operator class 3:16:2020 has started to get ball rolling again. I am still working on temp assignment for campaign disinfect all busses every 24 hours at mother clara hale depot. Some of the permit folks have went to training center for DMV road test.
I think as long as this virus/disease does not flare up as it horribly as it was they will START getting people in for final processing in July.