Hamiltonian Flows and Heisenberg picture of quantum mechanics

Question

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial)

In particular what I know is that:

1. the exponentiation of the Hamiltonian $H$ yields (the adjoint of) the propagator operator $U=e^{\frac{itH}{h}}$
2. To any operator $A$ we can associate its evolved $A_t=U^+AU$ so that given an initial state $\psi_0$ the average of the expected value of A at time t is $<A>_t=(\psi_0,A_t\psi_0)$ and it holds the Heisenberg equation $\frac{d}{dt}A_t=\frac{1}{ih}[A_t,H]+\partial_t A_t$.
3. The exponantiation of operators other than the Hamiltonian, such as angular momemtum L, momentum p yields propagations in'dimensions' other than time, in the sense that if $T=e^{\frac{itL}{h}}$ then $T\psi(\rho,\phi,\theta)=\psi(\rho,\phi,\theta+t)$ (where L has been expressed in z-axis spherical coordinates). Analogously $ T=e^{\frac{itp}{h}}$ then $T\psi(x)=\psi(x+t)$.

Now, in Hamiltonian mechanics we define Hamiltonian Flows as the group action $\phi_t$ associated to the solution $(x(t),p(t))$ of the 'generalized' Hamilton equations $$\frac{dx_k}{dt}=\partial_{p_k}F$$ $$\frac{dp_k}{dt}=-\partial_{x_k}F$$ so that $\phi_t(x(0),p(0))=(x(t),p(t))$.

$F$ may be the Hamiltonian, in which case we have time evolution of the system, but it may also be an affine function of position, so that we have (negative) momentum evolution (in the sense that $\phi_t(x,p)=(x,p-t)$) or of momentum and we have spatial evolution (in the sense that $\phi_t(x,p)=(x+t,p)$). If finally F equals L we have rotations.

Now, I cannot fully and deeply explain, if it is possible, the parrallelism I notice. In particular:

1. Can we identify in some sense the exponentiation of operators with the group action (of $\mathbb{S}^1$?) thus connecting the two scenarios?
2. What is the precise notion of 'dimensions' other than time, if any, in which the system evolves?
3. In the Schroedinger picture, Ehrenfest theorem allows to find an analog to the Hamilton equations, is this possible also in the Heisenber picture (it may be through the Heisenberg Equation , bu I cannot make this precise), in particular, also in the cases $F\neq H$?