Courtesy runner
 

NFHS rules
Do the NFHS rules specifically state that courtesy runner for the pitcher or catcher must remain the same for the entire game? Please cite NFHS rule.

NO. See Rule 8, Section 9. It's pretty well explained there.

Can you use a different CR for the same player in the same inning? For example, pitcher leads off inning getting a hit, CR number 11 goes in for pitcher. The team bats around, pitcher comes up again in the same inning- can you use CR number 22 for the pitcher?

Same answer as on the other forum. As long as the first CR is no longer on base and a bat-around would assure that; the original position player or another CR can run.
The only restriction within and inning is a player used as a CR can not be a sub in that inning except for an injury that would otherwise force shorthanded play.

Yes. While the rule is not completely clear on this, (using the phrase that a CR is "designated for that half inning" for example in 8-9-6), it seems that 8-9-6 is intending to make illegal a) bringing F1/F2 back in to run and b) bringing in another CR while the original is still on base. Each time F1 or F2 reach base, the team is allowed to designate a CR for that player.