Yes. We can divide through by one of the functions on the left-hand side and also assume $F=G$, and then the question becomes: Can $1+|h(z)|^2$ be the square of the absolute value of a holomorphic function, for a non-constant holomorphic $h$?

For this to happen, we need $\log (1+|h(z)|^2)$ to be harmonic. If you now just work out the Laplacian of this function and use the Cauchy-Riemann equations for $h$, you'll find that the numerator of the expression you obtain from two applications of the quotient rule equals $4((h_1)_x^2+(h_2)_x^2)$, so $h$ has to be constant.

Yes, for Question 1.

We can divide through by one of the functions on the left-hand side and also assume $F=G$, and then the question becomes: Can $1+|h(z)|^2$ be the square of the absolute value of a holomorphic function, for a non-constant holomorphic $h$?

For this to happen, we need $\log (1+|h(z)|^2)$ to be harmonic. If you now just work out the Laplacian of this function and use the Cauchy-Riemann equations for $h$, you'll find that the numerator of the expression you obtain from two applications of the quotient rule equals $4((h_1)_x^2+(h_2)_x^2)$, so $h$ has to be constant.

Yes. We can divide through by one of the functions on the left-hand side and also assume $F=G$, and then the question becomes: Can $1+|h(z)|^2$ be the square of the absolute value of a holomorphic function, for a non-constant holomorphic $h$?

For this to happen, we need $\log (1+|h(z)|^2)$ to be harmonic (where non-zero). If you now just work out the Laplacian of this function and use the Cauchy-Riemann equations for $h$, you'll find that the numerator of the expression you obtain from two applications of the quotient rule equals $4((h_1)_x^2+(h_2)_x^2)$, so $h$ has to be constant.

Yes. We can divide through by one of the functions on the left-hand side and also assume $F=G$, and then the question becomes: Can $1+|h(z)|^2$ be the square of the absolute value of a holomorphic function, for a non-constant holomorphic $h$?

For this to happen, we need $\log (1+|h(z)|^2)$ to be harmonic. If you now just work out the Laplacian of this function and use the Cauchy-Riemann equations for $h$, you'll find that the numerator of the expression you obtain from two applications of the quotient rule equals $4((h_1)_x^2+(h_2)_x^2)$, so $h$ has to be constant.