Where to stay in Santo Domingo

Zona Colonial is a good area to stay that has AirBnBs.

I would stay closer to where you need to go for these appointments.  Bella Vista, El Million, Naco are more central. You can find many air bnb listings there

If you are coming for extra time and want to explore then the Zona Colonial is a lot more interesting and you can walk to many places

I'm a few steps behind you so I haven't done any detailed research on this yet, but fwiw, I did notice that there are a lot of decent-seeming Airbnbs within walking distance of Lishali's office. Which looked like a pretty decent part of town to hang out in for a few weeks. Does anybody who actually knows SD have any thoughts about that?

A few weeks?  So if you are planning to visit for a few weeks I would not stay near Lishali's office. There is nothing wrong with that area at all if its for  2 or 3 days to process your residencia. But for a few weeks I would definitely want more within walking distance. For that duration please consider the Colonial Zone. There is so much to do in that  area!

No, not to visit -- to wait for the paperwork to go through.

Just word of advice as would not wait for paperwork go thru but rather until approved. Terrible process and they have been asking me for one new document every week or so instead of everything all at once like a normal process would.

Exactly you do not wait around for that to happen. Go to where you live here or where you want to be and wait.  Sometimes it can take months and months.

Approval takes months, not weeks!