What should I change about my setup? Do I need some sort of controller?Ubiquiti makes a controller, just that it's software based. (UniFi Controller) It's a free download. I think it also still requires Java (just a forewarning)
The house is a mid-century home with almost no attic access and I don't want to run cords up walls. I've found that I can place the access points in the basement with the dome side pointing up (beaming up instead of down).Have you considered using HomePlug to help with avoiding running wiring where possible?
Does Ubiquiti make a router that I should get to replace my nighthawk router?They make a Security Gateway. Never used it, so I'm not going to pitch its being any better or worse.
What about the access points -- should I get a stronger model?Let's at least take a crack with your current UniFi units first.
I understood the Ubiquiti products as geared toward "mesh networking."They do make some mesh products. However, what you chose is not one of them.
As a separate note, I'd not connect through the WAN port. Unless you want to protect the wireless devices from the wired ones, set the LAN side up on the same subnet, disable, DHCP, and connect one of the LAN ports to the main switch.