  Thin films of the high-temperature superconductor
HgBa$_2$CaCu$_2$O$_{6+\delta}$ have been prepared on SrTiO$_3$ substrates by
pulsed-laser deposition of precursor films and subsequent annealing in
mercury-vapor atmosphere. The microstructural properties of such films can vary
considerably and have been analyzed by x-ray diffraction and atomic force
microscopy. Whereas the resistivity is significantly enhanced in samples with
coarse-grained structure, the Hall effect shows little variation. This
disparity is discussed based on models for transport properties in granular
materials. We find that, despite of the morphological variation, all samples
have similar superconducting properties. The critical temperatures $T_c \sim
121.2$ K $\dots 122.0$ K, resistivity, and Hall data indicate that the samples
are optimally doped. The analyses of superconducting order parameter
fluctuations in zero and finite magnetic fields yield the in-plane $\xi_{ab}(0)
\sim 2.3$ nm $\dots 2.8$ nm and out-of-plane $\xi_{c}(0) \sim 0.17$ nm $ \dots
0.24$ nm Ginzburg-Landau coherence lengths at zero temperature. Hall
measurements provide estimates of carrier scattering defects in the normal
state and vortex pinning properties in the superconducting state inside the
grains.
