= new reply since forum marked as read
with a bovine avatar. 
You are stepping into a field of cow patty landmines.
I guess that your proposed test might give some indication as to whether the fart-passer ate at any ethnic restaurants if the gas chromatography detects a sufficient concentration of various spices. However, I don't want to be the lab tech that has to collect the sample.
I guess that I know why. 
I did take a course in adsorption spectroscopy in college and while I can not say definitively answer, my hunch is that the answer is yes. Although the THC should metabolize into various organic compounds there would also be sufficient amounts of residual cannabinoids that could be detected with the right instruments, particularly if using GCMS (gas chromatography/mass spectroscopy). The mass spectroscopy portion of the test would provide the molecular weight of the residual cannabinoids.
Here is an article that might be of interest that supports my reasoning and also provides some answers to your more generic questions:
http://chemistry.about.com/od/medicalhealth/f/What-Is-The-Chemical-Composition-Of-Farts.htm
It would be difficult to obtain, store, transport and test a gaseous specimen. A blood test for cannabis is pretty cheap. A urine test is even cheaper.
Any diagnostic information gleaned from a fart-test could probably be obtained from the history of present illness alone.
And without the bad smell...
Don't they now have smell machines - machines that can detect odors and analyze them? Yes, they do - or will soon!
We're getting closer to the day when your smartphone knows you have a cold before you do
By Randy Rieland
smithsonian.com
January 30, 2013
To mimic smelling, tiny sensors would be integrated into smartphones or other mobile devices and, as a breathalyzer can determine alcohol levels, they would gather data from the smell of your breath by detecting chemicals that humans wouldn’t perceive and send it to a computer in your doctor’s office. The thinking is that eventually this would be a core component of home health care–the ability to “smell” diseases remotely, such as liver or kidney ailments, asthma or diabetes.
Or on a more basic level, as IBM’s Hendrik Hamann put it: “Your phone might know you have a cold before you do.”
IBM is also working with health care organizations to equip patient and operating rooms with sensors that can help address one of the biggest problems hospitals face today–how do you keep them hygienic? Hundreds of sensors will basically sniff for cleanliness, identifying the chemical compounds that create odors, some of which are undetectable by humans. The staff can say they cleaned a room; the sensors will know if and when they did.
More: http://www.smithsonianmag.com/innovation/these-machines-will-be-able-to-detect-smells-your-own-nose-cannot-8790482/?no-ist
So what if they make a machine that can detect illicit drugs and set it loose to wander around your workplace, a school, public sidewalks or transportation? And if it detects a smell it doesn't like, it alerts the authorities - or takes a picture like a red light camera - and sets them after the emitter of the smell?
Talk about Big Brother intruding everywhere!
Only slightly
Did THC help inspire this question?