1. Baby oil or mineral oil should be just fine. I wouldn’t recommend any with extra fragrance though.
2. Fragrance is ok, just avoid other additives like aloe and such. I use Johnson & Johnson brand, plain mineral oil, and fragrance.
3. Yep, like others have said, just use regular mineral oil without additives (check the label). I use Johnson’s Baby Oil as it’s not much more expensive than knock-offs and I know what I’m getting. Go easy on the oil, though! A little goes a long way.
When your doll arrives, just give her a good cleaning to get all the factory muck off, followed by a light powder/cornstarch dusting once she is thoroughly dry. That’s really all you need to do in the beginning. When it comes time for an oil rub, don’t overdo it, just enough to coat the skin and let it absorb, (that can take several hours), before powdering. Generally, it’s wash, dry, oil (if it’s time for that) then powder.
4. Bathe them upon arrival, Vaseline crease areas as needed (elbows, underarms, groin, and knees), which depends on how often you move those areas and can be determined by a touch test. As for re-oiling, it depends on the TPE blend. Do a touch test. If it comes back slightly greasy, you are ok for a while longer. If not, it needs a redo.