Let $(X_\alpha, \mathcal{L}_\alpha)_{\alpha\in A}=(M, L, J_\alpha, \overline{\partial}_\alpha)_{\alpha\in A}$ be a family of polarized varieties, with $A$ the complex manifold parametrizing the family (assumed to be connected). Here is $M$ the common underlying smooth manifold of the varieties $X_\alpha$, $J_\alpha$ are the integrable almost-complex structures corresponding to the different varieties $X_\alpha$, $L$ is the common underlying smooth complex line bundle of the $J_\alpha$-holomorphic line bundles $\mathcal{L}_\alpha$, and $\overline{\partial}_{\alpha}$ are the integrable delbar-operators on $L$ corresponding to the different holomorphic line bundles $\mathcal{L}_\alpha$.
Lemma: For each $\alpha_0\in A$, there is a neighbourhood $U\subseteq A$ of $\alpha_0$, and a smoothly varying family $(\omega_\alpha)_{\alpha\in U}$ of symplectic forms in $c_1(L)$, such that for each $\alpha$, the form $\omega_\alpha$ is $J_\alpha$-Kähler.
Sketch proof: For each $\alpha$ we have a map
$$F_\alpha: \{\text{hermitian metrics on $L$}\}\to \{\text{real 2-forms in $c_1(L)$}\}$$ given by sending a hermitian metric on $L$ to $-i$ times the curvature of its $(J_\alpha,\overline{\partial}_\alpha)$-Chern connection. The maps $F_\alpha$ vary smoothly with $\alpha$. If $\omega_0$ is a $J_{\alpha_0}$-Kähler form, there is some hermitian metric $h$ on $L$ such that $F_{\alpha_0}(h)=\omega_0$. For $\alpha$ sufficiently close to $\alpha_0$, the $J_\alpha-(1,1)$-form $\omega_\alpha:=F_\alpha(h)$ is also nondegenerate, and so is $J_\alpha$-Kähler. $\square$
It follows that the space of 2-forms in $c_1(L)\in H^2(M,\mathbb{Z})$ which are Kähler for some $J_\alpha$ is connected. By the Moser trick, all these 2-forms are isotopic to each other. This answers your first question.
It also follows that one can smoothly select elements $(\omega_\alpha)_{\alpha\in a}$ of $c_1(L)\in H^2(M,\mathbb{Z})$, which are respectively $J_\alpha$-Kähler.
Edit: For the second question, I had previously proposed the pullback $\pi_1^*\omega_\alpha+\pi_2^*\widetilde\omega$ as a potential symplectic form on $M\times A$, where $\widetilde\omega$ is some symplectic form on $A$. But, now that I think about it, this pullback is not necessarily closed.
On the other hand, by the arguments for the first question, we can select smoothly varying diffeomorphisms $\psi:A\to\operatorname{Diff}(M)$ such that for all $\alpha$, $\psi_\alpha^*\omega_\alpha=\omega_{\alpha_0}$. (Maybe this requires simple connectedness of $A$?) Write $\Psi:M\times A\to M\times A$ for the induced diffeomorphism $\Psi(x,\alpha)=(\psi_\alpha(x),\alpha)$. Then $(\Psi^{-1})^*(\pi_1^*\omega_{\alpha_0}+\pi_2^*\widetilde\omega)$ is a symplectic form which restricts on $\alpha$-slices to $\omega_\alpha$, which should answer the second question.
I had also mentioned some general references on moduli spaces of Fano varieties (because of the issue of finding a symplectic form $\widetilde\omega$ on $A$), which I will preserve: 1, 2.
