Is the language $\{ 0^m 0^n : 0 \leq m \leq n\}$ regular?
Some people can be absolutely sure, and at the same time absolutely wrong. In this case, you are one of those people. 

A language is a set of strings. A language is not the rules that were used to express the language. $0^*$ and $\{0^m 0^n: 0 ≤ m ≤ n\}$ are the same language, even when that language is expressed in two different ways. And $0^*$ and $\{0^m 0^n: m ≥ n ≥ 0\}$ is again the exact same language, just expressed differently.

Your attempted "pumping lemma" proof fails for the same reason: For example, $0^5 0^4$ is in the language, because it is exactly the same string as $000000000$, which is again the exact same string as $0^0 0^9$.