In Avatar, Earth Benders can bend rock and sand, but would it be possible to bend wood and would it count as Earth?
Simple Answer: No. Earthbenders cannot bend wood because it lacks a mineral content. The exception to this might be petrified wood since it is wood that has been fossilized and now composed completely of stone.
The Avatar wiki gives us an extended answer and source under Weaknesses:
Metal and wood: An extension of the last weakness, because nearly all earthbenders do not know how to bend processed metals, the bending abilities of an earthbender can be negated by securing them within a metallic barrier, separating them from any contact with the earth. Toph, however, as the inventor of metalbending, is an exception to this. However, even metalbenders cannot bend highly purified metals such as platinum. Wood cannot be subject to earthbending because it is neither a mineral nor mineral-like.
Hamilton, Joshua (writer) & Volpe, Giancarlo (director). (November 2, 2007). "The Runaway". Avatar: The Last Airbender. Season 3. Episode 7. Nickelodeon.