What you should do is buy some Kilz Latex Primer/Sealer. Buy a 3/8" knap roll for your roller. Put it on heavy. After it has dried (the next day is best), apply your top coat of paint. Make sure you buy a paint that has a mildewcide already mixed into it. (This type of paint is designed for rooms with high humidity, like a bathroom.) Use a 3/8" Finish Roller for this job. Be prepared for a 2nd coat.
You see, the Primer/Sealer will prevent the Pink from showing through the White. You may want to consider 2 coats of Primer/Sealer especially if it's dark. Yes, I'd sand the glossy surface first. It promotes adhesion.
As far as the Pink door is concerned, you can stip off the paint. However, you'll need to refinish the door in its entirety so it's uniform on all sides. Weight this 'work' with the cost of simply buying a replacement. (You don't necessarily have to buy a pre-hung door.) In either case, since you're working on a bathroom, you want to protect the door somehow. Talk to the folks at the Paint Store (found in the Yellow Pages under PAINT - RETAIL) about options. I suggest you DON'T use the paints in the Home Centers. IMO, they're just not that good. Check out an FYI I posted yesterday: New Paint Group On The Block