Yes to all. A little belt skip is reasonable depending on what turn motor you may be running. If it's constant under gentle acceleration or if you're just using drag brake...then there might be a problem. And most definitely yes to cleaning all the old belt pieces out of the pulleys. I've actually had to pry pieces out of used chassis using a dental pick. When the belts are old , they not only fall apart but they kind of melt and make a real mess.
NF...all done with touring cars of any kind.