Yes you can use an alkyd enamel. I recommend you reprime the items with a fast drying oil based primer (Kilz Oil Base) and lightly sand for a smooth finish. This applies to smooth trim and doors. If your doors are embossed, wood grained, you can skip the repriming.

Two coats of paint is a minimum amount. The actual amount of coats needed depends on your color. You could use a fast drying oil based primer tinted to your finish color for the first coat.