Hey good job getting the old gal going.
On the volume issue, if you're located in a very strong signal area, the radio is actually working normally. In the early days, stations were generally weaker and more distant, and reception was on a long wire outside antenna. Many designers put the volume control right at the antenna input.
But today with nearby powerhouse stations, the signal gets into the set's wiring and RF tubes 'downstream', bypassing the front end vol.control. That's why you can't get the volume down to zero. The set's owner should see this as part of the charm of using such a vintage radio in today's signal environment.
To "fix" the problem would entail some judicious RF shielding of the radio.
For your own info, there's some good tutorials on neutrodyne sets if you google "neutralizing the neutrodyne".