  We use new deep 21 cm HI observations of the moderately inclined galaxy NGC
4559 in the HALOGAS survey to investigate the properties of extra-planar gas.
We use TiRiFiC to construct simulated data cubes to match the HI observations.
We find that a thick disk component of scale height $\sim\,2\,\mathrm{kpc}$,
characterized by a negative vertical gradient in its rotation velocity (lag) of
$\sim13 \pm 5$ km s$^{-1}$ kpc$^{-1}$ is an adequate fit to extra-planar gas
features. The tilted ring models also present evidence for a decrease in the
magnitude of the lag outside of $R_{25}$, and a radial inflow of $\sim 10$ km
s$^{-1}$. We extracted lagging extra-planar gas through Gaussian velocity
profile fitting. From both the 3D models and and extraction analyses we
conclude that $\sim10-20\%$ of the total {\HI} mass is extra-planar. Most of
the extra-planar gas is spatially coincident with regions of star formation in
spiral arms, as traced by H$\alpha$ and GALEX FUV images, so it is likely due
to star formation processes driving a galactic fountain. We also find the
signature of a filament of a kinematically "forbidden" HI, containing $\sim
1.4\times 10^{6}$ M$_{\odot}$ of HI, and discuss its potential relationship to
a nearby HI hole. We discover a previously undetected dwarf galaxy in HI
located $\sim 0.4^{\circ}$ ($\sim 58$ kpc) from the center of NGC 4559,
containing $\sim 4\times10^{5}$ M$_{\odot}$. This dwarf has counterpart sources
in SDSS with spectra typical of HII regions, and we conclude it is two merging
blue compact dwarf galaxies.
