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We present a photometric study of the dwarf galaxy population in the core region ($< r_{\rm vir}/4$) of the Fornax galaxy cluster based on deep $u'g'i'$ photometry from the Next Generation Fornax Cluster Survey. All imaging data were obtained with the Dark Energy Camera mounted on the 4-meter Blanco telescope at the Cerro-Tololo Interamerican Observatory. We identify 258 dwarf galaxy candidates with luminosities $-17 < M_{g'} < -8$ mag, corresponding to typical stellar masses of $9.5\gtrsim \log{\cal M}_{\star}/M_\odot \gtrsim 5.5$, reaching $\sim\!3$ mag deeper in point-source luminosity and $\sim\!4$ mag deeper in surface-brightness sensitivity compared to the classic Fornax Cluster Catalog. Morphological analysis shows that surface-brightness profiles are well represented by single-component S��rsic models with average S��rsic indices of $\langle n\rangle_{u',g',i'}=(0.78-0.83) \pm 0.02$, and average effective radii of $\langle r_e\rangle_{u',g',i'}\!=(0.67-0.70) \pm 0.02$ kpc. Color-magnitude relations indicate a flattening of the galaxy red sequence at faint galaxy luminosities, similar to the one recently discovered in the Virgo cluster. A comparison with population synthesis models and the galaxy mass-metallicity relation reveals that the average faint dwarf galaxy is likely older than ~5 Gyr. We study galaxy scaling relations between stellar mass, effective radius, and stellar mass surface density over a stellar mass range covering six orders of magnitude. We find that over the sampled stellar mass range several distinct mechanisms of galaxy mass assembly can be identified: i) dwarf galaxies assemble mass inside the half-mass radius up to $\log{\cal M}_{\star}$ ~8.0, ii) isometric mass assembly in the range $8.0 < \log{\cal M}_{\star}/M_\odot < 10.5$, and iii) massive galaxies assemble stellar mass predominantly in their halos at $\log{\cal M}_{\star}$ ~10.5 and above., 19 pages, 9 figures, accepted for publication in ApJ