I use medium grit regular sandpaper and have always had good results. For drywall sanding, I use a flat hand sander with special drywall sanding screens that don't clog up the sanding surface.
Drywall compound is the "mud" drywallers use to cover seams, patch holes, etc, in drywall installation. You can get a small four pound pail or so pre-mixed. It is thinner than spackle, and spreads easier over thin uneven surfaces than spackle. Some deep patches (it doesn't sound like you have any, though) require more than one mud application.
I like to use kitchen/bath semi-gloss rather than eggshell. Some of the newer paints like this also have extra or special mold/moisture prevention chemicals in it, as I recall. This paint scrubs better over time, and in smaller bathrooms, light colors can help the available light make the room seem brighter. Have used Valspar over the years with good success. Consumer Reports put out some pretty good test results on interior paints. Test results don't always show that more expensive paints are best values.