Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172).
Note that Tian's theorem states that all del Pezzo surfaces which are not a blow-up of $\mathbb{P}^{2}$ in 1 or 2 points have a KE-metric, but these come in trivial families.
By Moser's trick the Kähler forms on each variety of the family are abstractly symplectomorphic (see here symplectic form on an algebraic family), but they cannot be identified biholomorphically, whilst simultaneously preserving the symplectic structure. This provides a negative answer to question 1.