Re: original trim -- how to clean up?
Clean and degrease the trim first, with a heavy-duty cleaner, and a scrubby/sponge. TSP, spic & span, Fantastik, etc. work. Since it's already alligatoring, I'm not too worried about cleaning too vigorously, but if the finish were intact, you'd want to be more gentle.
When it's clean and dry, you can start sanding with 220-grit sandpaper and sanding sponges.
Don't sand through the varnish, but sand down to the alligatoring.
There's a possibility that the original finish was shellac (a high possibility), but tha alligatoring was caused by re-coating (years later) with some kind of oil finish. The two layers bond imperfectly, and the upper layer crawls on the previous coat as is dries out. This is alligatoring.
The situation will require a little experimentation to get a satisfactory finish. And if it looks good to begin with, it may not last many years.
Things you can try for varnish:
Sealcoat Shellac. De-waxed. Dries very clear and will be glossy unless rubbed out with steel wool and wax. May craze the underlying finish.
Regular shellac, 3 lb. cut: Contains wax, do not use.
Wipe-on gel varnish: Satin finish. May cloud/haze over old finish. Durable, has some polyurethane in the mix. Not as shiny as shellac. Worth a try, IMO.
Brush-on poly: To be avoided, as it can't easily be removed, would have same if not worse clouding risk.
Acrylic poly: Never tried it in this situation, but perhaps worth the experiment.
Wipe-on oil finishes:not for over-coating existing finishes, ever.
Remove not the ancient landmark, which your fathers have set.