We study general properties of the Landau-gauge Gribov ghost form factor $\ensuremath{\sigma}({p}^{2})$ for $\mathrm{SU}({N}_{c})$ Yang-Mills theories in the $d$-dimensional case. We find a qualitatively different behavior for $d=3$, 4 with respect to the $d=2$ case. In particular, considering any (sufficiently regular) gluon propagator $\mathcal{D}({p}^{2})$ and the one-loop-corrected ghost propagator, we prove in the $2d$ case that the function $\ensuremath{\sigma}({p}^{2})$ blows up in the infrared limit $p\ensuremath{\rightarrow}0$ as $\ensuremath{-}\mathcal{D}(0)\mathrm{ln}({p}^{2})$. Thus, for $d=2$, the no-pole condition $\ensuremath{\sigma}({p}^{2})<1$ (for ${p}^{2}>0$) can be satisfied only if the gluon propagator vanishes at zero momentum, that is, $\mathcal{D}(0)=0$. On the contrary, in $d=3$ and 4, $\ensuremath{\sigma}({p}^{2})$ is finite also if $\mathcal{D}(0)>0$. The same results are obtained by evaluating the ghost propagator $\mathcal{G}({p}^{2})$ explicitly at one loop, using fitting forms for $\mathcal{D}({p}^{2})$ that describe well the numerical data of the gluon propagator in two, three and four space-time dimensions in the SU(2) case. These evaluations also show that, if one considers the coupling constant ${g}^{2}$ as a free parameter, the ghost propagator admits a one-parameter family of behaviors ( labeled by ${g}^{2}$), in agreement with previous works by Boucaud et al. In this case the condition $\ensuremath{\sigma}(0)\ensuremath{\le}1$ implies ${g}^{2}\ensuremath{\le}{g}_{c}^{2}$, where ${g}_{c}^{2}$ is a ``critical'' value. Moreover, a freelike ghost propagator in the infrared limit is obtained for any value of ${g}^{2}$ smaller than ${g}_{c}^{2}$, while for ${g}^{2}={g}_{c}^{2}$ one finds an infrared-enhanced ghost propagator. Finally, we analyze the Dyson-Schwinger equation for $\ensuremath{\sigma}({p}^{2})$ and show that, for infrared-finite ghost-gluon vertices, one can bound the ghost form factor $\ensuremath{\sigma}({p}^{2})$. Using these bounds we find again that only in the $d=2$ case does one need to impose $\mathcal{D}(0)=0$ in order to satisfy the no-pole condition. The $d=2$ result is also supported by an analysis of the Dyson-Schwinger equation using a spectral representation for the ghost propagator. Thus, if the no-pole condition is imposed, solving the $d=2$ Dyson-Schwinger equations cannot lead to a massive behavior for the gluon propagator. These results apply to any Gribov copy inside the so-called first Gribov horizon; i.e., the $2d$ result $\mathcal{D}(0)=0$ is not affected by Gribov noise. These findings are also in agreement with lattice data.