We present a comprehensive high spatial resolution imaging study of globular clusters (GCs) in NGC 1399, the central giant elliptical cD galaxy in the Fornax galaxy cluster, conducted with the Advanced Camera for Surveys (ACS) aboard the Hubble Space Telescope (HST). Using a novel technique to construct drizzled point-spread function libraries for HST/ACS data, we accurately determine the fidelity of GC structural parameter measurements from detailed artificial star cluster experiments and show the superior robustness of the GC half-light radius, r(h), compared with other GC structural parameters, such as King core and tidal radius. The measurement of r(h) for the major fraction of the NGC 1399 GC system reveals a trend of increasing r(h) versus galactocentric distance, R-gal, out to about 10 kpc and a flat relation beyond. This trend is very similar for blue and red GCs, which are found to have a mean size ratio of r(h),(red)/r(h),(blue) = 0.82 +/- 0.11 at all galactocentric radii from the core regions of the galaxy out to similar to 40 kpc. This suggests that the size difference between blue and red GCs is due to internal mechanisms related to the evolution of their constituent stellar populations. Modeling the mass density profile of NGC 1399 shows that additional external dynamical mechanisms are required to limit the GC size in the galaxy halo regions to r(h) approximate to 2 pc. We suggest that this may be realized by an exotic GC orbit distribution function, an extended dark matter halo, and/or tidal stress induced by the increased stochasticity in the dwarf halo substructure at larger galactocentric distances. We compare our results with the GC r(h) distribution functions in various galaxies and find that the fraction of extended GCs with r(h) >= 5 pc is systematically larger in late-type galaxies compared with GC systems in early-type galaxies. This is likely due to the dynamically more violent evolution of early-type galaxies. We match our GC r(h) measurements with radial velocity data from the literature and split the resulting sample at the median r(h) value into compact and extended GCs. We find that compact GCs show a significantly smaller line-of-sight velocity dispersion, = 225 +/- 25 km s(-1), than their extended counterparts, = 317 +/- 21 km s(-1). Considering the weaker statistical correlation in the GC r(h) color and the GC r(h)-R-gal relations, the more significant GC size-dynamics relation appears to be astrophysically more relevant and hints at the dominant influence of the GC orbit distribution function on the evolution of GC structural parameters.