Can I paint over my glossy white cabinets?

If they are painted with high-gloss paint, you do need to sand them down.

The door doesn't need to be sanded, but if it's bare wood it'll need a coat of primer before you paint it - this will cut down on the number of coats of paint you need to apply. You should use primer meant to go on stained wood (by "stain", I mean wood stain, not dirt stain). If you don't use a stainblocking primer, the stain will eventually migrate through the paint and give the door a blotchy appearance.

You may also need to plane the edge of the door before painting. Paint has a definite thickness; two or three coats of paint can make your door too tight in the frame to close properly. I have this in the new home I just bought and I'm being forced to plane and repaint all the doors because someone put three coats of cheap paint on them without planing.

Unfortunately, yes, you need to sand and prime the cabinets. If the glossy paint is an oil base paint, the latex paint is not going to stick to it. If you are not sure, go with an oil base paint. It will go over latex without any problems.

The door, probably not. The cabinets...definately sand them first. It will be worth the time in the end. You might want to buy a mini palm sander from Lowes or Home Depot. They are pretty cheap and make your job a snap.

Yes, you will need to sand and primer. The latex paint might chip and scratch easily. For all of your hard labor, you might want to consider an oil base paint. Good luck!