$p(0)\geq c'(0)$ in monopoly
You intuition about the assumption is correct. 

Given the usual assumptions of downward sloping demand curve ($p'(q)&lt;0$) and non-decreasing supply curve ($c''(q)\ge0$), if $p(0)&lt;c'(0)$, then demand would lie entirely below supply. This means the equilibrium quantity must be zero---as if the market did not exist. Therefore, for the analysis to be non-trivial, the assumption of $p(0)\ge c'(0)$ is maintained. 

With the use of calculus, we are assuming implicitly that quantities are infinitely divisible. Thus $c'(0)$ can be roughly interpreted as the cost of producing the "first infinitesimal unit" of the good. Mathematically, 
\begin{equation}
c'(0)=\frac{\mathrm d\, TC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,(FC+VC(q))}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,VC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}.
\end{equation}
Thus $c'(0)$ is the derivative of the variable cost function, evaluated at $q=0$.