Acrylic paints should stick just fine to plastic. Usually when people struggle with painting steel doors, it is because they are not keeping a wet edge. You have to break the door down into sections and when you are working in one section, you can't really get paint outside of that area and then brush back through it minutes later. If you brush through paint that is already starting to set, that is when you will pull at it which will look like hell. Do your panels first and keep the paint in the panels. Then do your horizontals, then the left and right sides. You may want to switch to using a small roller if you struggle with getting the door to look good by painting it with a brush.

I painted the door today. I'm waiting for it to dry before I close it, but here are before and after pictures while it was open. Before. The spot above the lock is where I used rubbing alcohol to check if was latex paint on the door. The color difference of hardware is when the ex walked out, I changed the locks, and I didn't want to change the door lever also. After:

Nope. Unless you want it to be a feature or focal point in your livingroom. Just paint it the same as the closet doors. And if you want to make it a bit more interesting, you could add a wood woven blind if that look would go with the rest of the room.

Those are great doors, a lot nicer style than your old ones! I would stain the outside, even if it's a little more maintenance, because it would look so much better. It's typical for doors to be painted white on the inside, and I don't think it would look cheap, especially if youl are careful to use the same color and sheen as the rest of your woodwork.