Is there any code, standard, reference, or documentation that states how many times a circuit breaker can be cycled safely after it has tripped? Also, are there a set number of times a circuit breaker can trip before its overcurrent properties begin to degrade?

Its not in the code but the MFG should be able to advise you.

A safe answer is once! Really, I don't know..... Good question!

I guess it depends on if the CB opened on overload or on short circuit.
Some larger CBs are expensive, and assuming the internals are compromised after one or two overcurrent conditions could get expensive.
There are companies that test CBs.
Maybe that would be the way to go if you are talking about larger commercial / industrial systems.

It is a good question.