Not currently, though I am considering it. Right now I host mine on a VPS in linode. Though i need to downgrade it, I built it with the expectationg of allowing joins, but recently decided just to keep it private.

Can you just choose any free name for the domain if it’s on your own hardware or do you need to rent one regardless?

This wouldnt work. You not only need to have a routable/real domain name, but the server likely needs access to the internet to allow fro federation, specifically ingress traffic, to work.

Do you keep it active all the time or turn it off for the night/other periods of time where you know you won’t use it?

Mine runs 24/7. Even if i hosted it at home it would be 24/7. Only issue is

I already use port 80/443 at home. So i would need to reconfigure NGINX to use a proxy, which could also break federation. I could do that, in fact I am pretty sure the ansible config uses NGINX proxy commands, just that I would have to customize it and Im lazy. I already have stuff on VPS systems in linode (blog, teamspeak etc) so its no biggy to have another one.
My internet at home can be flaky. For example I currently dont have power at home and while I normally run on UPS for a time, and can cut to generator when I am home, my network just went into auto-shutdown.