These are style vs. practicality questions in my opinion, and that means you're going to have to duke this out with your hubby. Sounds like you have opposing tastes, and that, my friend, I think is going to be your biggest problem.
I have tile in my master bath and slate in my guest bath. They are both functional, clean up well, and look gorgeous. I think it is a total matter of taste...what look are you going for? Slate is more rustic and often more masculine looking, and tile more crisp and feminine. What's the decor of your home? Maybe use that to determine which way to go.
As for wainscoting...only looks good in older homes IMO. So if you have a Victorian for instance it's a nice compliment to wallpaper prints for instance. A tile floor might look nice with that look as well. No to wainscoting and slate. Yucky-poo.
If you paint the door white on one side and leave the wood alone on the other side, it looks fine. We did this in an older home (Victorian) we had once. It's a bear trying to keep the white from bleeding on the edges, but if you tape, you should manage to get a crisp, uniform look.
If you do tile, you'll definitely want it to match or compliment the floor where color theme is concerned! So white tile floor ...you'll want white tile walls. You can add flair with borders or intermingle printed tiles for instance in a pattern throughout the wall. Having one wall tiled may work, but it all depends on shape of the room, whether there is a window or not, is it behind a toilet (might look like a urinal you know) or on a solid wall with nothing around it. Once again..that's a style thing.