Difference between Hamiltonian in classical Mechanics and in quantum Mechanics

$\bullet$ The wave function has information about both position and momentum, so in a sense you're right. But it's not particularly useful to count quantities in the way you do here: Write $X_\Psi:=(q,p),\ \nabla:=(\tfrac{\partial}{\partial q},\tfrac{\partial}{\partial p})$ and $\omega:=\begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$. Now your classical equation is $\dot X_\Psi=\omega \nabla H$, where $H$ and therefore also the vector $\omega \nabla H$ is a function of $X_\Psi$, and this is also just one equation for one quantity.

$\bullet$ Yeah, the wave function is viewed as a weighted deviation for the action principle for classical point particles. But for the one-particle wave function, the Schrödinger equation is just a field equation and this also has a Lagrangian, $\frac{\partial \mathcal{L}}{\partial \psi^{*}} - \left(\frac{\partial}{\partial t} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{*}}{\partial t}} + \sum_{j=1}^3 \frac{\partial}{\partial x_j} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{*}}{\partial x_j}}\right) = 0$ with $\mathcal{L}\left(\psi, \mathbf{\nabla}\psi, \dot{\psi}\right) := \mathrm i\hbar\, \frac{1}{2} (\psi^{*}\dot{\psi}-\dot{\psi^{*}}\psi) - \frac{\hbar^2}{2m} \mathbf{\nabla}\psi^{*} \mathbf{\nabla}\psi - V( \mathbf{r},t)\,\psi^{*}\psi$.

$\bullet$ These are the definitions, right. But both provide both, energy function an operator generating time developement. The function for the Hermitian operator maps $\psi$ to $\left\langle\phi\right|H\left|\phi\right\rangle$, see here, and the equations above provide a flow mapping a state at a time $t_i$ to a state at a time $t_f$.