Author here: the docker project for the tile server allows setting up automatic syncing fairly easily:

https://github.com/Overv/openstreetmap-tile-server#enabling-...

I'm not sure about Nominatim or Valhalla's ability to auto-sync the latest changes, at least with the dockerized version, but it may be possible.

I think the official Nominatim server has minutely updates so I'd be very surprised if the open source version isn't the same and can't do that as well. I've never set it up though, so I'm not sure where to find those docs.

Yes, you can do minutely updates with Nominatim, instructions are here: https://nominatim.org/release-docs/latest/admin/Update/