It is possible to copy and paste some of the database elements between FMP12 and earlier versions, but there will be enough <missing> and <unknown> messages from the process to make a LOT of work patching things up. I've spent some time consolidating some of our separate databases into a single database and it is hard work even if you are using the same version of Filemaker for both. You'd need a good database documenting tool such as Base Elements or InspectorPro to have any chance of success. Not for the faint-hearted.
Both the G4 and FMP10 are in my view at the end of their lives, so any work done for them is going to have a very limited payback. The advantage of rewriting the database is that you should be able to streamline it and get read of all the old clutter that seems to accumulate in databases over time, so hopefully it will be smaller and faster. Personally, if I had time to rewrite a database, I'd do it in the latest version of Filemaker.
Buy a new Mac Mini, install FMP12 on it and put the old G4 out to grass; then you can do the rewrite at your leisure.