Quasi-steady state hypothesis application
Thanks for the interesting question.  Before we get into the details, let me note a few things:


Your system is a model of substrate inhibition, a phenomenon where higher-than-optimal concentrations of a substrate for an enzyme decrease its own rate of conversion to product.  
It sounds like you have a good handle on the enzyme mass balance, i.e. that $e_0 = e + c_1 + c_2$ is a conserved quantity whose time derivative is everywhere zero (no approximations or hypotheses required) in this kinetic scheme.  This makes sense because enzymes are catalysts that accelerate reactions without any net participation in them.


OK, onward to your questions.


The assumption of fast equilibrium for $c_2$ is a different assumption than the PSSH hypothesis.  Assuming a fast equilibrium for $c_2$ means that reactions 3 and 4 happen much more quickly than reactions 1 or 2.  Thus the concentration of $c_2$ is determined solely by the thermodynamics of reactions 3 and 4: $$k_4 c_2 = k_3 s c_1 \;\;\; \Rightarrow  \;\;\; c_2 = \frac{k_3}{k_4} s c_1$$
The pseudo-steady-state hypothesis could be applied to $c_1$.  $$\frac{
dc_1}{
dt} = k_1se_0 - (k_1s+k_2+k_3s)c_1+(k_4-k_1s)c_2 = 0$$  Substituting in the result of the fast equilibrium assumption (point #3) gives $$k_1se_0 - (k_1s+k_2+k_3s)c_1+(k_4-k_1s)\frac{k_3}{k_4}s c_1 = 0$$  You can solve this equation for $c_1$ and plug the result into your $\frac{
ds}{
dt}$ equation.
If you are interested in understanding more about the mathematical validity of the PSSH, you should know that it is related to singular perturbation.  For a treatment of a "simple" Michaelis-Menten system by singular perturbation, see this book excerpt.  The PSSH solution is an "outer" singular perturbation solution.