If one does not assume $X$ and $X'$ compact, then there are relatively easy counter-examples: there is a vector bundle over an elliptic curve with total space analytically equivalent to $\mathbb{C}^\*\times\mathbb{C}^\*$, see e.g., Peters-Steenbrink, Mixed Hodge structures, p. 102.

However, if I think when $X$ and $X'$ are (possibly singular) algebraic surfaces, the answer to question 2 is positive. (As far as I remember, this was proved by J. Steenbrink; If I find a precise reference, I'll post it.)

If one does not assume $X$ and $X'$ compact, then there are relatively easy counter-examples: there is a vector bundle over an elliptic curve with total space analytically equivalent to $\mathbb{C}^\*\times\mathbb{C}^\*$, see e.g., Peters-Steenbrink, Mixed Hodge structures, p. 102.

There are positive results as well, but the only one I can think of is this: if $X$ and $X'$ are (possibly singular) compact algebraic surfaces, the answer to question 2 is positive, see Steenbrink-Stevens, Indag. Math., 46, 1984, no. 1, p. 63-76.