Not sure how to state this without it being taken wrong, but does it have a set screw? Is it loose? IF not, it needs to be loosened.
How old is your lathe? IF it is a year or two old, it should come loose easily; if it is 3 years or more, dust and grim build up could be causing it.
2. Spray with some liquid wrench (or WD-40) and let it sit for a couple of hours.
3. I'm not the one to recommend tapping or not tapping. I am mechanically inclined as I grew up around motors, pulleys, sprockets, shafts motors etc. What is light tapping to one person - is hard tapping to others. I don't know how hard or light "cautiously tap" is to you. I would tap stuck pulleys on one side and go around tapping in different spots until I could see it moving even a small amount. Then back and forth. For me, instead of tapping with a hammer, I use a flat ended punch that doesn't make dents and put it to the pulley and tap it. Go around and tap it in different places until it comes loose.