Is P equal to the intersection of all superpolynomial time classes?
Yes. 

In fact, by the McCreight-Meyer Union Theorem (Theorem 5.5 of McCreight and Meyer, 1969, free version here) a result of that I believe is due to Manuel Blum, there is a single function $f$ such that $\mathsf{P} = \mathsf{DTIME}(f(n))$. This function is necessarily superpolynomial, but "just barely."

The theorem applies more generally to any Blum complexity measure $\Phi$ and any union class $\bigcup_{f \in S} \mathsf{BLUM}_{\Phi}(f(n))$ where $S$ is a c.e., self bounded set of total computable functions. (A set of functions $S$ is c.e. if there is a single partial computable function $F(i,\vec{x})$ such that $S = \{f_i(\vec{x}) | i \in \mathbb{N}\}$ where $f_i(\vec{x}) := F(i,\vec{x})$. Self-bounded means that for every finite subset $S_0 \subset S$, there is a function in $S$ that dominates all $g \in S_0$ almost everywhere. "$\mathsf{BLUM}_{\Phi}$" is a notation I haven't seen before, but I like it :) - I'm using it for the $\Phi$-bounded analog of a time-bounded complexity class.)