We present a detailed multi-wavelength study (from rest-frame UV to far-IR) of narrow-band (NB) selected, star-forming (SF) H$\alpha$ emitters (HAEs) at $z \sim 2.23$ taken from the High Redshift(Z) Emission Line Survey (HiZELS). We find that HAEs have similar SED-derived properties and colors to $sBzK$ galaxies and probe a well-defined portion of the SF population at $z \sim 2$. This is not true for Ly$\alpha$ emitters (LAEs), which are strongly biased towards blue, less massive galaxies (missing a significant percentage of the SF population). Combining our H$\alpha$ observations with matched, existing Ly$\alpha$ data we determine that the Ly$\alpha$ escape fraction ($f_{\rm esc}$) is low (only $\sim$ 4.5\% of HAEs show Ly$\alpha$ emission) and decreases with increasing dust attenuation, UV continuum slope, stellar mass, and star formation rate (SFR). This suggests that Ly$\alpha$ preferentially escapes from blue galaxies with low dust attenuation. However, a small population of red and massive LAEs is also present in agreement with previous works. This indicates that dust and Ly$\alpha$ are not mutually exclusive. Using different and completely independent measures of the total SFR we show that the H$\alpha$ emission is an excellent tracer of star formation at $z \sim 2$ with deviations typically lower than 0.3 dex for individual galaxies. We find that the slope and zero-point of the HAE main-sequence (MS) at $z \sim 2$ strongly depend on the dust correction method used to recover SFR, although they are consistent with previous works when similar assumptions are made.