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.

I had also mentioned some general references on moduli spaces of Fano varieties, which I will preserve: 1, 2.

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.

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. If, say, $A$ admits a Kähler metric $\widetilde\omega$, then the pullback $\pi_1^*\omega_\alpha+\pi_2^*\widetilde\omega$ on $M\times A$ is, I think?, a Kähler metric for the complex structure on the family. This reduces your second question to the question of whether the moduli space is naturally Kähler, which could be proved, for example, that showing it is projective. For this you should ask the experts on the kind of moduli space you are interested in, for example, in the case of Fano varieties, these folks or these folks.

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.

I had also mentioned some general references on moduli spaces of Fano varieties, which I will preserve: 1, 2.