Polynomial Hierarchy --- polynomial time TM
Your definition of $\Sigma_2^P$ is a bit misleading. Both $u_1$ and $u_2$ have to be bounded in length by some polynomial in $|x|$, and this usually forms part of the formula rather than part of the descriptive text. The predicate $M$ has to be polynomial time in the combined input length, which is the same as being polynomial time in $|x|$ since both $u_1$ and $u_2$ have length polynomial in $|x|$.

You have not described the predicate SubgraphIsomorphic correctly. You are given two graphs $A,B$, and you have to decide whether $B$ is isomorphic to a subgraph of $A$. We write it as an NP-predicate as follows:
$$(A,B) \in \mathrm{SubgraphIsomorphic} \leftrightarrow \exists |P| \leq |(A,B)| \mathrm{Isomorphic}(A,B,P), $$
where $\mathrm{Isomorphic}(A,B,P)$ is true if $P$ encodes an injection from the vertex set of $B$ to the vertex set of $A$, and $(x,y)$ is an edge of $B$ iff $(P(x),P(y))$ is an edge of $A$. This is a polytime predicate. Moreover, the injection can be coded in length polynomial in the input $(A,B)$ (for simplicity, I'm assuming that the encoding is such that $|P| \leq |(A,B)|$).