Are there any tests for halitosis?

Yes, but not needed. Halitosis (bad breath) after certain foods or drinks occurs for several reasons. Chemicals from some foods/drinks diffuse into the flesh in the mouth and throat. These chemicals (odors) keep coming out into the breath for many hours. Some foods/drinks cause odors when in the stomach. Some substances stick to the teeth. Special tests for "gases" are not needed, as the odor is usually obvious.
Might be. We typically don't run a test for this. I have seen some sort of a meter that is supposed to measure sulfur, but i cannot say if it is actually accurate or meaningful. Typically if you have excellent oral hygiene (twice per day) and regular gum cleanings, and no periodontal disease and don't smoke, etc, halitosis can be easily controlled. Sugarless gum can be helpful too.
Yes, certainly. There are several tests have been developed for halitosis. There is device called "halimeter", or breath analyser, also gas chromatography, and several other tests, that mesure bacteria produced gases, known as vsc or bacterial by-products, as cadaverin. We can pretty much test and mesure halitosis precisely.