What do I need to research to switch to an internally geared hub?
The first thing you should be looking into is gearing to ensure that an internally geared hub will work for you! The more you pay, the more gear range you get out of a hub, but even the nicest internally geared hubs don't match the spread of standard mountain bike gearing.
If you've decided that an Alfine 11 hub is for you then to answer your questions in order:
-the axle size shouldn't be an issue.
-the hub spacing is 135mm. That's mountain bike spacing typically. Apparently it can be adapted down to 130mm but I'm not going to detail how to do that here
-sell it/keep it/ use it as a paperweight
-The hub will have a dedicated shifter. I seem to remember that there are both trigger and grip shift options. Should fit without issue, though you'll probably have to take your grips/bar tape off.
-Yes, unless you have an incredibly nice rim with the proper drilling that you want to re-use.
You will also need a chain tensioner if your bike does not have horizontal/track-style dropouts, which I doubt it does. I believe there's one that goes along with the Alfine that Shimano sells as an accessory, though I don't think you have to have that particular one.