Let $T$ be a linear algebraic torus over $\mathbb C$ and $X$ be a smooth quasi-projective symplectic $T$-variety. Also, assume that the action of $T $ is free and $X/T$ exists as a smooth variety. Is this action Hamiltonian?

If the above action is Hamiltonian let us denote the algebraic moment map $X\rightarrow t^{\vee}$ by $\mu$. Does there exist a one-to-one correspondence, via $\mu$, between the symplectic leaves of $X/T$ and the coadjoint orbits of $T$ (which are just points of $t^{\vee}$ in this case)?

This is an affirmative answer to $1$ (assuming by Hamiltonian you mean in the sense of real symplectic geometry).

Yes. Due to Sumihiro's theorem (+ standard facts about Hamiltonians). In fact the assumptions smooth quotient and symplectic are not necessary, just smooth and quasi-projective is enough. Throughtout when I mean symplectic I mean in the real sense.

Sumihiro's Theorem ("Equivariant completion", J. Math. Kyoto Univ., 14: 1–28) implies that: For any normal, quasi projective variety with an algebraic $T$-action there exists $N>0$ and an $T$-equivariant embedding $$\phi : X \rightarrow \mathbb{CP}^N,$$ for some algebraic $T$-action on $\mathbb{CP}^N$.

Next, it is known that every algebraic torus action on projective space is linear, so in a particular homogeneous coordinate system of $\mathbb{CP}^N$ the action is given by $$z.[z_{0}: \ldots : z_{n}] = [z^{a_{0}}z_0 : \ldots : z^{a_{n}}z_n] , $$ for some integers $a_{0}, \ldots a_n$.

Then, this action is Hamiltonian (with respect to the Fubini-Study Kahler form on $\mathbb{CP}^n$) with Hamiltonian $$ H([z_{0}: \ldots : z_{n}] ) = \frac{\sum a_i |z_i|^2}{\sum|z_i|^2}.$$

Finally, there is a general fact about Hamiltonian torus actions, the proof is linear algebra.

Fact: Let $(M,\omega)$ be a symplectic manifold with a Hamiltonian $T$-action and moment map $F$. Then if $N \subset M$ is an invariant, symplectic submanifold then the restricted action on $N$ is Hamiltonian with moment map $F|_{N}$.

So applying this result to the symplectic submanifold $\phi(X)$ to obtain that the action is Hamiltonian. I could probably find references for these statements if you'd like.