Will I be able to match the paint on peeling cabinets reasonably well?
Not perfectly, but if you make good choices about which panels to paint it won't be conspicuous. 
Yes, but I'd limit sanding to the rough edges of the paint left behind after scraping the loose material away. If your cabinets aren't solid wood you risk damaging the veneer and creating a fuzzy appearance.
Absolutely. That's the only reliable way to get a good paint bond. Use a high-quality primer with excellent adhesion. 
That will depend on you. Follow common best practices and it will look just fine. You can also use a foam brush if you don't want any grain at all.