Hint:
we can solve such kind of problems using coordinates. Let the cube : $$ A=(0,0,0) \quad B=(a,0,0) \quad C=(a,a,0) \quad D=(0,a,0) $$ $$ E=(0,0,a) \quad F=(a,0,a) \quad G=(a,a,a) \quad H=(0,a,a) $$ Now we can find any distance using the formulas of analytic geometry.
As an example, for your question 1. :
the first step is to find the equation of the plane passing thorough $BEG$.
Use $BE=(-a,0,a)$ and $BG=(0,a,a)$ as vectors on the plane and find the orthogonal vector $\vec n= BE \times BG= (a^2,a^2,a^2)$, so the equation of the plane orthogonal to this vector and passing thorough $B$ is $(a^2,a^2,a^2) \cdot(x-a,y,z)=0$ i.e. $$ a^2x+a^2y+a^2z-a^3=0 $$ and the distance of $F$ from this plane is: $$ d=\dfrac{a^3+a^3-a^3}{\sqrt{a^4+a^4+a^4}}=\frac{a}{\sqrt 3} $$ (see here )
In the same way you can find the answer to the second question.