The enchanting fixes seem to have worked, at least the duration one. It's 1440 now.
Something went wrong? Do I have to do something AFTER I install it? I am also running MGE (installed before Code Patch) and I'm adding and removing mods every now and then.
I did NOT enable the Showcase plugin included with the MWCP. Should I have? I was happy enough with the changes being described to me.
1440 is the default duration, the patch (if you apply it) reduces it to 300. So it seems that either applying the patch didn't work (in which case you should've gotten an error message), or you're starting a different exe than the patched one. Do you have multiple Morrowind installs and start Morrowind with the Launcher? That won't work since the Launcher always starts the exe in the path that's stored in the registry. Try to start Morrowind.exe directly, see if the changes are there now. If not, re-apply the patch.
You don't need to do anything after applying the code patch. MGE is compatible. The showcase mod does not need to be enabled, it's just for demonstration and to offer a quick opportunity to test whether the patch worked.