The wording of the extended question makes this feel like a philosophy of mathematics question, rather than the more obvious one about Peano arithmetic.
The whole point of mathematics is this:
Starting from the (stated) assumption that certain things are so, can we convince ourselves beyond any reasonable doubt that other things must also necessarily be so?
Pure mathematics makes no statement about whether the initial assumptions (axioms) actually hold in the real world. Mathematics just makes statements about statements, in an attempt to convince you of the truth of other statements.
If you are not convinced by the proof, then there is more work for the proof to do.
If enough other people are convinced by the proof, then possibly the gap is within your own understanding of the proof – or perhaps you have identified a flaw in the original proof, which you may be able to persuade others to accept.
In order to help you with the step you are currently struggling with, $0 + 0 = 0$, it would help us to know where you started your journey, and what it is safe for us to assume.
Personally, I'd consider the existence of $0$ to be axiomatic (i.e. it exists because we all agree that we have to assume something in order to get started with anything†), and the statement $0 + 0 = 0$ to be true, because it is part of the definition of the + operator.
Does that mean that $0$ exists in the real world? Mathematicians don't really care about that. Physicists, engineers and accountants might, and indeed do, because it permits them to make a living building bridges that don't fall down, but it isn't a question that Pure Mathematicians can really answer.
† Maybe there is room for some mathematics even more fundamental than this, where even simpler assumptions enable us to deduce the existence of $0$, but I don't know what it is...