It seems like you're asking two questions. Correct me if I'm misunderstanding.
1. How to deal with the chipped paint. I don't see primer under that chipped paint. I wonder if they painted right over the wood? Anyway, scrap away anything remotely loose. Sometimes you can't tell until you run a scraper over the paint and suddenly paint you thought was adhered just breaks off. So scrap every inch of those jambs. Sand, prime, light sand, paint x 2 (as mentioned above).
2. The stained part of the jamb needs to be painted white. Use a shellac based primer (stir well before use) to cover up the brown. 2 coats to be safe, sand, paint x 2.
I won't tell you which paint to use, but I will say I've not been happy using Sherwin's new Emerald Urethane trim paint. It just doesn't level well (or at all it seems). I suspect a pro painter would find a way to work with it, but I tried with and without Floetrol, good brushes, mohair rollers, foam rollers. With Floetrol and a roller worked the best, but I won't be using this on trim because it's not leveling well.