Is this the correct way to use the pumping lemma?
A subset of a regular language could be non-regular, so the entire line of reasoning is flawed. As an example, every non-regular language over $\{0,1\}$ is a subset of the regular language $(0+1)^*$.

Regarding the language $H$, you'd do better writing your proof in paragraph-style rather than as a cryptic list. In addition, you made the following mistakes:


You never mention which $h$ you take.
You don't get to choose the decomposition $h = xyz$. The proof has to work for every such decomposition satisfying the properties given in the lemma ($|xy| \leq n$, where $n$ is the "pumping constant").


Several ways of showing that your original language is not regular are given in this answer.