mac mini as backup

Yes in a way. You can set the Mac mini into Target Disk mode, and connect it via a cable to the new Mac mini.

See here: link-> Transfer files between two Mac computers using target disk mode - Apple Support

Once connected toehold Mac should appear as hard drive in Finder on the new Mac.

You can then export the photos the old Mac hard drive as a type of "backup".

A much easier and more flexible way would be to connect the old MacMini to your network and enable file sharing on it.

FWIW, I have an older 2010 Mini that I added a few external drives to and it now serves as a home media/file server. Set up as such I can access info from all my Macs and Apple media content via Home Sharing on my iDevices.

No. You can set it up to Target Disk Mode regardless of macOS version.

If you are using iCloud Photo Library, you can turn it on on the new Mac and it will downlaod the Photos onto itself.

From there you can export them onto the old Mac mini. Not sure you can turn on iCloud Photos on Mavericks though someone will correct me If I’m wrong.