Regarding boiling chamber reside, if you boil the initial charge down to about 5-10% the initial volume and then pour off the remainder you can reduce the accumulation. A periodic soak with white vinegar should take care of the rest of it. You can also get commercial solutions for this, since this is a common problem with tea kettles.
If you add a water line, remember to dump the chamber periodically or all the minerals that are left behind will build up and reduce your heating efficiency faster.
Increasing the speed of distillation may impact the quality of your result. Running faster will, in principle, carry over somewhat more non-water chemicals. However, you'd have to test to see if that matters. I'm guessing you won't notice a difference.
Assuming you are boiling a gallon of 60F water with a household circuit limited to about 1500W at best it's going to take around an hour unless you can get some significant heat recovery from condensation going. Maybe you could add a peltier to pump heat from the condenser to the boiling chamber (no idea if running a peltier with a hot end at 100C is going to be efficient enough to bother with).
I don't think doing it under vacuum will help very much, since most of your heat energy (600Wh or so) is going into the phase change rather than temperature change.
You might want to experiment with adding small amounts of salts to the distilled water to adjust the flavor, and aerate it to put dissolved gasses back in. Evidently these can have a large impact on the flavor.