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 https://mathoverflow.net/questions/284809/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.