I remember hearing that trees and other plants actually obtain a large amount of their mass from the carbon floating in the air, not the ground beneath them. Does the makeup of air actually contain enough carbon to support this theory, and is a tree's surface area actually large enough to obtain the amount of carbon it needs directly from the air?
The vast majority of a tree's carbon comes from the air, which averages 0.03-0.04% by volume (300-400 ppmv) CO2. This is fixed through photosynthesis and eventually stored as glucose which the plant can then use for its metabolism.
Doing some quick math, this means that in order to produce 1 kilogram of carbohydrates (e.g. cellulose) a plant needs to process on the order of 2000-3000 cubic meters of air (and ≈550 g or mL of H2O), which would fill a cube measuring 13-14 meters on a side. Note this is an ideal figure; a plant's fixing efficiency will likely fall as it depletes the air of CO2.
Plants do take a great deal from the ground, namely water, fixed nitrogen (for proteins), phosphorous (for nucleic acids), and several ions (sodium, potassium, calcium, among others)
I'm surprised nobody has mentioned Jan van Helmont. To summarise Blankenship's account, in the 17th century, he (Helmont, not Blankenship) grew a tree in a known dry weight of soil and weighed the fallen leaves of the tree, and then eventually the whole tree including the root system. He found that the mass of the soil had barely diminished and concluded that the tree's mass was overwhelmingly due to watering (CO2 had not been discovered.)1
With the benefit of knowledge of CO2 we can conclude that the mass of trees comes from the fixation of CO2 and H2O as well as the presence of H2O as a biological solvent.
(1) Blankenship, R. E., Molecular Mechanisms of Photosynthesis, Blackwell Science, 2002, p. 26
Yes. In fact the organic compounds' mass comes mostly from the air, since Photosynthesis essentially builds up glucose by only adding hydrogen to CO₂. The 2 H₂O → 2H₂ + O₂ reaction can be treated seperately, as was determined by Sam Ruben and Michael Kamen with ¹⁸O isotope tracing, i.e. in fact only the hydrogen in the carbohydrates comes from the soil, and this has of course a much smaller mass.
As said by Nick T, the more complex compounds also incorporate other elements that the plants get from the soil, but most of them still consist mainly of C and O. The total mass of a tree of course also has a lot of water and some minerals in it, but it's still safe to say that a tree "consists mainly of air".
To your question about surface area and air CO₂ content, neither is usually the limiting factor: in nature, light exposure, water and mineral supply are (although in enclosed spaces like greenhouses CO2 fertilization is sometimes necessary).