Why do algebraic proofs apply to cryptography? How do we know that the number theoretic and algebraic results used in cryptography provide a perfect model for the behavior of integers as implemented in computers? Does there exist a bijection between the purely mathematical integers and the two’s complement integers, and are subsets of the two’s complement integers under the modulus operator isomorphic to the corresponding (mathematical) integers mod n?

If so, are these conditions sufficient for all cryptographic proofs relying on algebra and number theory?
Integer operations as implemented on computers are isomorphic to a theoretical definition of integers. Otherwise operations would not give the correct results.

Given the terminology in your question, I suspect that when you think of integer operations on a computer, you're thinking of operations on machine words. Cryptography uses numbers that don't fit in machine words: any cryptography code that works on integers (i.e. generally speaking asymmetric cryptography) must include a bignum library (or rely on a third-party one).

Two's complement is a way to represent negative integers. It's irrelevant to cryptography because cryptography pretty much never uses negative integers: only nonnegative integers ($\mathbb{N}$, or integers modulo $n$ which are stored using their representative in the range $[0,n-1]$.

You can find a formal proof of the equivalence between Peano arithmetic and arithmetic on binary representations in many places, for example in the BitNat module of Coq.

Implementing integer operations is a non-problem as far as getting the correct result, but it does have an impact on security. If the implementation is not careful, side channels such as timing and memory access patterns may leak confidential data that compromises the security of a protocol that would be secure if the adversary could not find out any information about the intermediate values.