Yes, you can litter train one outside of it's cage. Infact many people don't even cage their bunnies. Alot of rabbits get ful-run of the house. It's alot easier to start out by keeping the rabbit in the cage with the litter box for the first few weeks. The litter box is best put in a corner as that is where bunnies usually like to go.

Yes, lop-eared rabbits are more prone to ear infections because it is a warm, moist environment, exactly like with dogs. I believe that a lop-eared rabbit needs to have his ears cleaned on a regular basis by a vet, though I'm not too sure on the details of that.

I, myself, am allergic to rabbits...so I know what you mean about the itchiness lol. It's always my nose that gets itchy though. But I want to have a flemish giant rabbit one day anyways. I'm allergic to cats and dogs and have those, might as well get a rabbit. Maybe that's one way to wear the allergies down, who knows!