= Solution
Use the <Hamiltonian vector field> convention $\iota_{X_{H_t}}\omega=-dH_t$. A smooth time-dependent <Hamiltonian function> $H:[0,1]\times M\to\mathbb R$ with flow defined over the whole interval generates the <Hamiltonian isotopy> determined by
$$
\frac d{dt}\phi_H^t=X_{H_t}\circ\phi_H^t,\qquad \phi_H^0=\mathrm{id}.
$$
The <group of Hamiltonian diffeomorphisms> is
$$
\boxed{\operatorname{Ham}(M,\omega)=\{\phi_H^1:H\text{ generates such a flow}\}}.
$$
These maps are <symplectomorphisms>: by <Cartan's magic formula>, $\mathcal L_{X_{H_t}}\omega=d\iota_{X_{H_t}}\omega=0$. Thus $d(\phi_H^t)^*\omega/dt=0$.
The term “group” is justified explicitly. If $\phi_t,\psi_t$ are generated by $H_t,K_t$, their product path is generated by $H_t+K_t\circ\phi_t^{-1}$, since the pushforward of $X_{K_t}$ by a <symplectomorphism> is $X_{K_t\circ\phi_t^{-1}}$. The inverse path is generated by $-H_t\circ\phi_t$. The zero Hamiltonian gives the identity.
On a closed <symplectic manifold>, all these flows exist on the finite time interval. On an open manifold one often means the <compactly supported Hamiltonian diffeomorphism group> $\operatorname{Ham}_c$, using Hamiltonians with compact support in $[0,1]\times M$. We use that subgroup for the finite-energy definitions on Euclidean space below. The unrestricted complete-flow group is needed for the global cotangent translations in Question 4; those translations are generally not compactly supported.
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