If you plan to take the Chuo line to Otsuki, you might prefer to stay near Shinjuku. OTOH, if you stay in Higashi Nihonbashi, you might find it easier to reach Mt. Fuji by via Shinkansen from Tokyo Station and renting a car from, say, Mishima.
I hope it shows enough of the region for your purposes. Your other option is to get a local map (in Japanese) and pore over it with the English map on the side.
Not a bad plan. It's such a small town - you're in the country and away from difficult city driving. It's quite a pretty drive from there through the valley up to Kawaguchiko. You might also consider taking your camera up the hills along that route (I mean parking the car and climbing on foot) for the views they afford of Fuji and the lakes.
If you're after sunrise photos, perhaps you should consider staying by Lake Yamanaka. It seems to offer the most spectacular sunrise views of the mountain. (For a sunset shot with Fuji lit up red (Beni Fuji), you ought to be near Shojiko or Motosuko).
It's a small town and I never had a problem finding parking - even on festival days when it was packed. English language skills are worse than you'll find in many European countries - but levels of patience are pretty good. You'll find people willing to invest a lot of effort to help you. If you're not confident about where you parked, ask the receptionist to look and confirm it's OK.

I don't think that Fujiyoshida has an awful lot to recommend it, but the home-style Udon restaurants are worth checking out. You'll need to ask a local where to find them. Unless you've a keen eye for noticing things like the over-sized kitchen for preparing the noodles, you're likely to walk right past them, thinking that they're ordinary homes.