ive been using ublue’s ucore server for about a year without any real complaints. Fedora coreos would also work fine as a base if you wanted to not use ublue’s opinionated ucore.
you could also checkout bootc-image-builder and make your own, on your own hardware so you’re not relying on github.
Sure! You could use systemd timer or cron if you want to automate builds.
Edit: This looks like it may be helpful if you’re wanting to host everything yourself - https://www.youtube.com/watch?v=5ZN_7NDvavY