Question about Load voltage when triac is switch off
I think your reverse engineering is in error. C101 and R103 would be a snubber directly across MT1 and MT2 and usually shown to the right of the triac. R102 would usually be connected directly to MT1.

Figure 1. A typical opto-isolated circuit. Image unattributed on Triac Switching circuit with Optocoupler

Should I be getting 0V across the motor when the opto triac is off?

With such a small load the snubber will pass enough current to give that voltage across the load.

If yes, would a gate resistor be a good idea and if so, should I connect that to neutral?

I don't think it will make any difference.

Is it likely that the 7V on the motor is causing that hum? If not, is it possible to say what might be causing it?

The leakage current is likely to be the cause of the hum and otherwise shouldn't cause any problems.

Links

Phase Control Using Thyristors by Littlefuse.

Thyristor Theory and Design Considerations Handbook
 by ON Semiconductor.