The leather strip on my arrow rest is all but worn out.  Lot's of arrows have flown off it over the years, but I'm not sure how normal this is.  Does this happen to anyone else?  Do I need to increase brace height or make some other modification?

Arrow rests will need to be replaced periodically. Sounds like you have gotten a lot of use out of it. I would guess your set up is just fine, it's just time for a new rest.