Thanks much for the kind words, and welcome!
When it comes to home command station cores, those can't get plugged back into anything -- but they are nicely mobile, so that's helpful. IIRC, you got an AIP bump from losing the actual home command station in the first place, and it just keeps happening if you lose more of these. Best thing is to get up a defensive perimeter so that hopefully you don't lose any more home planets. Granted, though, it seems like in my 3/4 player games we only have 1-2 home stations left by the time we finish. The GOOD thing about the home cores is that they actually give you more resources, so your friend is economically better off now.
In terms of the database of all the ships, you can see that sort of thing in-game by pressing the F1 key to cycle through the list of ships by mark. There's also a Reference in-game if you look in the STATS button window. And lastly, if you want to export the "excel secondaries" for perusal outside the game, then hit F3 on the main menu, then Ctrl+Shift+F8. That will close the game and write some files into a Data folder inside your RuntimeData folder.
Hope that helps!