Do I need to take weight of the rocket into account when calculating escape velocity? Here there is the old problem.
I know from the old problem that the work $W_v$ that I need to make a rocket fast  enough to reach the escape velocity is
$$W_v= G \frac{mM}{r}$$ 
therefore because $$W_v=F\cdot S = G \frac{mM}{r} \rightarrow F_v=\frac{W}{S}=G \frac{mM}{rS} $$
that is the force I need to make a rocket fast enough to reach the escape velocity BUT
Do I also have to count the weight of the rocket?
If yes then the equation will be like this:
$$F_f=F_g - F_v= G \frac{mM}{r^2}-G \frac{mM}{rS}=G \frac{mM}{r}\biggl(\frac{1}{r} \cdot \frac{1}{S}\biggr) = G \frac{mM}{r}(rS)^{-1}  $$
 A: Escape Velocity doesn't depend on mass(but work depends on mass) of a rocket, Escape Velocity equation looks like this:
$$
v=\sqrt{\frac{2GM_{earth}}{r}}
$$
And Work is just change in kinetic energy $\Delta KE$ or $KE_{final}-KE_{initial}$, So Work equation looks like this:
$$
W=\frac{m_{rocket}v_{final}^2}{2}-\frac{m_{rocket}v_{initial}^2}{2}=\frac{m_{rocket}}{2}(v_{final}^2-v_{initial}^2)=\frac{m_{rocket}\Delta v^2}{2}
$$
And Don't forget that Fuel mass is decreasing (When Engine is powered) So That means that acceleration is changing (It's just Newton's second law that says this: $a=\frac{F}{m(t)}$ or full equation looks like this: $a=\frac{F}{m_{rocket}+m_{fuel}(t)}$). In order to solve this you have to "construct" differential equation. (The aforementioned equation in differential form looks like this: $\ddot x= \frac{F}{m_{rocket}+m_{fuel}(t)}$ and $V=\int_{t_0}^{t_1} a(t)$ $dt$.
A: 
Do I also have to count the weight of the rocket?

You already have! This is the $m$ in $G\frac{m M}{r}$.
I don't understand the line of reasoning for your second equation, and you made a subtraction/multiplication error as DavePHD pointed out. The value $F_g+F_v$ would be the total acceleration experienced by astronauts (or the payload) sitting in your rocket, but I don't know what $F_g-F_v$ would mean if you're accelerating away from your gravitational source, and I don't know what you want it to mean. So I can't really help you on that one.
The force needed to reach escape velocity over a certain distance does depend on the object's mass. The velocity needed to escape a planet is independent of the mass of the rocket.
