Wrong. Try checking out Peano's axioms for natural numbers, or the meaning of "complete ordered field".Numbers can never have an entirely abstract basis.
Well, there are several things one could mean by "counting" -- if you simply mean naming the terms of the sequence 0, 1, 2, 3, ... in order, then it's fairly straightforward.How would you teach someone to count just using numbers?
Nope. They don't have to represent anything, and even when they do, it doesn't have to have anything to do with quantity. For example, integers can represent proofs of formal logic.They have to represent some sort of quantity whether its a unit of length, area, volume or apples.