We have BM Sag harbor gray doors and baseboard and walnut stained wood windows with BM Delaware Putty walls. We've had lots of compliments so far! The picture is not the best (a bit too dark), but perhaps it gives you some idea. Here is a link that might be useful:

"It seems like some kind of resin though perhaps they just oiled the wood (???)." Probably varnish. Depending on the exact formula the stuff can remain pretty soft and elastic (Spar varnishes) or harden up to very hard stuff. You can use a satin finish poly, or any of a number of varnish type finishes. #0000 steel wool will reduce the sheen while leaving a pleasing satin finish once the varnish (or poly) is hardened. Non-woven abrasives can also be used to rub out the finish to the desired degree of gloss. Satin finishes use addatives to achieve the desired gloss, but mechanical works fine also. Even shellac has been used to finish wood trim. Usually one of the darker shades. Since the trim is not normally touched on a regular basis a couple coats of shellac can actually make a nice finish that is easily repaired by cleaning with paint thinner, then applying another coat of shellac.

Gotta be careful with heat guns. I have one, and if the application I'm heating is attached to the house I use it on the low setting. Many a house has been burned down with a heat gun. There is the real potential it will blow a smoldering ember into a recessed area that eventually catches fire. I was talking about the quartz bulb infrared paint remover. Doesn't blow any potentially fire starting debris into nooks and crannies. I figured it might be an old apartment building. But I've seen old houses with that many layers of paint too. And always wondered when there was already too much paint on there, why even more was applied. It's not like it looks any better... Peel Away will get really expensive really fast. But, I cannot imagine heat stripping that much paint off of that much wood work. You'll need an assortment of different size scrapers and and arsenal of fresh blades. Because the first four layers of lead paint, the eight layers of non lead oil paint, and the 12 layers of latex paint will dull those blades fast. Before committing to peel away, try Citri Strip. It's a lot less expensive, and you can get it at Home Depot. From my experience it works pretty well.

Clean, prime with Zinsser 123 primer (or Zinsser BIN - I've used both but 123 is water based.) . Lightly scuff sand with 220 grit, clean dust, 2 coats of paint. I took the advice from the paint forum from Faron and Brushworks years back on primers, paint, and paint tools and I've used this for many, many paint projects. The only time I used the BIN was when I wanted to seal pine knot holes from bleeding through the paint later. Only if money were no object would I replace with MDF. I personally wouldn't bother to do so. Yes, MDF would paint nicer than oak because it doesn't have grain but painted grain doesn't make me twitch to look at and this is up at the ceiling so wouldn't be as noticeable anyhow.