We use new deep 21 cm H I observations of the moderately inclined galaxy NGC 4559 in the HALOGAS survey to investigate the properties of extraplanar gas. We use TiRiFiC to construct simulated data cubes to match the H I observations. We find that a thick-disk component of scale height ∼2 kpc, characterized by a negative vertical gradient in its rotation velocity (lag) of ∼13 ± 5 km s-1 kpc-1 is an adequate fit to extraplanar gas features. The tilted ring models also present evidence for a decrease in the magnitude of the lag outside R 25, and a radial inflow of ∼10 km s-1. We extracted lagging extraplanar gas through Gaussian velocity profile fitting. From both the 3D models and extraction analyses we conclude that ∼10%-20% of the total H I mass is extraplanar. Most of the extraplanar gas is spatially coincident with regions of star formation in spiral arms, as traced by Hα 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” H I feature, containing ∼1.4 × 106 M ⊙ of H I, and discuss its potential relationship to a nearby H I hole. We discover a previously undetected dwarf galaxy in H I located ∼0.°4 (∼58 kpc) from the center of NGC 4559, containing ∼4 × 105 M ⊙. This dwarf has counterpart sources in SDSS with spectra typical of H II regions, and we conclude that it is two merging blue compact dwarf galaxies.