Can you mix these two paints?

How can I tell what type of paint was used on bathroom cabinets, Latex or oil base?

We just bought a home and have been struggling with choosing a new color for our great room (kitchen/dining/living room/walls going up the stairs/hallway/office with open glass french doors upstairs) in this unique home in the woods. The previous owners had a country style, hence the yellow paint they chose

We are about to paint our house and need some advice. I think the previous owner was color blind. The house is a VERY bright turquoise color that I call blue-green algae. We plan to paint the house a subdued sage gray-green. The guys at Sherwin Williams are insisting that

Regular off white with two doors, made of melamine. Can I paint it? If so, what kind of paint should I use? Is there any prep work?