Is it possible to run a Tor Hidden Service with remote assets, i.e. files, databases, etc? Would such a setup decrease the security and anonymity of the service? Would the remote assets also need to be within hidden services?
For databases, etc.: It would probably make sense to either keep them on the same server, not allowing connections from Tor/the internet in general. Else, for example when you have a hidden service running on one server and another server with the DB it might make it possible/easier to correlate this traffic with your hidden service. You should at least make sure any such connections are encrypted, but you should do that either way, even when not using Tor.
However, if you create such a setup you really want to double check that there are only authorized connections possible. So for example use a VPN and make the database bind to the IP of that VPN only. As usual multiple layers of security (firewall, VPN, ACL, ..) might increase your security.