(Abridged) Carbon-enhanced metal-poor (CEMP) stars in the halo components of the Milky Way are explored, based on accurate determinations of the carbon-to-iron ([C/Fe]) abundance ratios and kinematic quantities for over 30000 calibration stars from the Sloan Digital Sky Survey (SDSS). Using our present criterion that low-metallicity stars exhibiting [C/Fe] ratios (carbonicity) in excess of [C/Fe]$ = +0.7$ are considered CEMP stars, the global frequency of CEMP stars in the halo system for feh $< -1.5$ is 8%; for feh $< -2.0$ it is 12%; for feh $<-2.5$ it is 20%. We also confirm a significant increase in the level of carbon enrichment with declining metallicity, growing from $<$[C/Fe]$>$ $sim +1.0$ at feh $= -1.5$ to $<$[C/Fe]$>$ $sim +1.7$ at feh $= -2.7$. The nature of the carbonicity distribution function (CarDF) changes dramatically with increasing distance above the Galactic plane, $|$Z$|$. For $|$Z$|$ $< 5$ kpc, relatively few CEMP stars are identified. For distances $|$Z$|$ $> 5$ kpc, the CarDF exhibits a strong tail towards high values, up to [C/Fe] $>$ +3.0. We also find a clear increase in the CEMP frequency with $|$Z$|$. For stars with $-2.0 <$ [Fe/H] $< -$1.5, the frequency grows from 5% at $|$Z$|$ $sim 2$ kpc to 10% at $|$Z$|$ $sim 10$ kpc. For stars with [Fe/H] $< -$2.0, the frequency grows from 8% at $|$Z$|$ $sim 2$ kpc to 25% at $|$Z$|$ $sim 10$ kpc. For stars with $-2.0 <$ [Fe/H] $< -$1.5, the mean carbonicity is $<$[C/Fe]$>$ $sim +1.0$ for 0 kpc $<$ $|$Z$|$ $<$ 10 kpc, with little dependence on $|$Z$|$; for [Fe/H] $< -$2.0, $<$[C/Fe]$>$ $sim +1.5$, again roughly independent of $|$Z$|$.
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