Guided Waves - Helmholtz Equation

I am having trouble understanding the Guided Fields derivation process. Firstly, We derive the Helmholtz equation easily:

$Δ⋅vec{E}+ frac{∂^2vec{E}}{∂z^2} – gamma$₀$⋅vec{E}= 0$

Then we decompose the Helmholtz equation into two new equations thanks to decomposing the field in tangential and transversal components ($vec{E}=vec{E_t}+vec{E_z}$):

$Δ_t⋅vec{E_t}+ frac{∂^2vec{E_t}}{∂z^2} – gamma$₀$⋅vec{E_t}= 0$

$Δ_t⋅vec{E_z}+ frac{∂^2vec{E_z}}{∂z^2} – gamma$₀$⋅vec{E_z}= 0$

The first thing I do not undestand is why we can apply variable separation to solve this differential equation, are not we leaving some solutions applying this method?

$E_z(x,y,z) = F_E(x,y)⋅e^{-gamma⋅z}$

$H_z(x,y,z) = F_H(x,y)⋅e^{-gamma⋅z}$

Secondly, the boundary conditions, which are derived from Maxwell’s equation and the boundary with a perfect conductor, are the following:

TE Mode: $frac{∂F_H}{∂n} = 0$

TM Mode: $F_E = 0$

How do we get the boundary conditions of the TE Mode?

And lastly having all this conditions and the Helmholtz equation, how we can assure that $k_c^2$ is real and greater than $0$

$Δ_t⋅F_{E/H}+ k_c^2⋅F_{E/H}= 0$ , $frac{∂F_H}{∂n} = 0$, $F_E = 0$

Thank you very much, any help is appreciated.