Does the field of organic chemistry cover all the other group 14 elements or does it strictly deal with Carbon chemistry? Since the other group 14 elements have similar chemical properties, such as silicon, wouldn't they fall under the same discipline? Why or why not?
The other group 14 elements don't have similar chemistry. In general, the elements in the 2p block are "unusual" compared to the 3p/4p/5p blocks, just as how the elements in the 3d block are quite different compared to the 4d/5d blocks, and just as how hydrogen ("1s block") is very different from the alkali metals.
The obvious difference in chemistry comes from the unparalleled ability of carbon to catenate, or in other words, to form chains of single bonds. Silicon does not do this to any appreciable extent, mainly due to bond energetics (the Si–Si bond, for example, is much weaker than C–C).
Many of the functional groups familiar to organic chemistry have little parallel in silicon chemistry. Carbonyl groups are ubiquitous, but Si=O compounds are barely stable (see Wikipedia), which is a result of the poor π overlap between Si (valence 3p orbitals) and O (valence 2p). The same applies for C=C and C≡C bonds. Silicon analogues are known, but are very difficult to make and generally need to be stabilised using a variety of tricks (see: How come there aren't any silicon analogs of alkenes or alkynes? and links within).
So, traditionally, the chemistry of Si/Ge/Sn/Pb is within the realm of inorganic chemistry. Of course, it is never a clear-cut distinction, since these elements can coexist with carbon in the same molecule. But that is a story for another time.