I was thinking about using my old phones with something similar, but I'm worried about battery. When permanently connected to charger, won't they swell, start fire or something like that after longer(er) period of time?

On some models, if you root them, you can cap charging to reduce the chances of that happening.

https://f-droid.org/en/packages/mattecarra.accapp/

With some phones you can set a charge limit in software at like 50%, that should pretty much solve it I believe.

You can also "just" take the battery out, it might still function without it or with something that pretends to be a battery but isn't