There are couple of factors at work when re-steeping green tea: temperature, time, and the quality of the tea (the size & way it's been processed). Green tea is supposed to be brewed with water that's been brought to boil and allowed to cool to 167 - 176 degrees Fahrenheit (though many people simply heat the water to that point or what they eyeball as hot but not boiling). Green tea is typically steeped for 1 - 2 minutes. And green tea, especially loose leaf, but also with higher quality tea bags, has furled and sometimes rolled leaves. The lower temperature, the short steeping time, and the curled leaves mean that flavor is still left within the leaves even after multiple steepings. The leaves continue to expand and release in subsequent baths. I don't, unfortunately, know any of the exact science--what's being released when, etc.
However, a possible alternative to boiling & brewing individual teas when done or simply using more tea (the typical choice when brewing larger quantities), you can bring your water to temperature, steep, pour it into a preheated or heat retaining container, and repeat the cycle until you've brewed your multiple cups in one go. As far as I know the tea leaves don't need to 'rest' in between brewings.
The reason green tea can be re-steeped is also the reason most black teas can't. Black tea is subject to boiling water for a longer period of time (typically Western use is 3 - 5 minutes; some cultures favor an even longer steep for the deeply bitter tannins). The hotter and longer steep pulls more flavor out of the leaves faster. Some black teas, especially high quality whole-leaf, can be re-steeped, however. Most bagged black teas--and a lot of loose leaf--can't.
The heat & steep time is dictated by the processing of the tea once it's picked. Green tea has minimal processing--pluck, wilt, shape & dry. Black tea is allowed complete oxidation (called fermentation). Fermentation breaks down chlorophyll, releases tannins, and forms many of the taste & aroma compounds that typify black tea. Oolongs and other varieties are subjected to varying degrees of fermentation. The size & shaping of the tea leaf--allowed to remain whole, broken, rolled, or (for most bagged tea) cut into fannings and dust, also affects how tea should be initially steeped and whether it can be re-steeped. Highly rolled oolongs, for example, often hold up to more steepings than green tea and are even said to taste best only on the 3rd or further steeping.
You can't (or at least I can't) tell without a lot of experience with teas of that sort. Generally, a sufficiently fresh tea will have a decently strong fragrance when you smell the dry leaves. As it grows stale, the fragrance will be replaced gradually by mustiness. But which fragrance to expect varies dramatically by the tea. For instance, pan-fried greens like dragonwell (longjing) tend to smell vaguely nutty, while a sencha will smell more grassy. Dragonwell will also be more yellowish while the sencha will be more greenish. But these are two of many different varieties of green.
Also, there's a continuum of oxidation from green to black, so "too oxidized" is really a matter of taste. If you buy a Darjeeling (black), it'll tend to be lightly oxidized for a black tea; if your green is darker than that, maybe it shouldn't be called green.
I'm sure with appropriate analytical chemistry equipment you could determine some aspects of tea "quality"--but this would be complete overkill unless you're running a major tea business and you want to ensure quality. And even then, you'd probably be better off hiring a number of taste-testers.
There is one test (but it's not definitive): brew the tea, then leave the cup out for a day. If it doesn't get dramatically darker by the end of the day, it's already mostly oxidized.
In any case, if you like the tea, why do you care so much about the quality? You can find some outrageously expensive teas--I've noticed some South Korean greens in particular--that start getting into the price ranges of fine wines, but unless you're doing this to impress people, you're better off trying a variety of samples and then getting the one/ones you like. (Hopefully you won't find that you really like those South Korean greens....)
Best Answer
I've never had an issue reusing anything that's real tea (black, oolong, green). Perhaps there is another factor here? If you're black tea is coming out of a tea bag, or a lower "grade" then the green tea leaves you're using that might be a reason that it isn't coming out desirably.
Generally speaking it's an issue of grade rather than the type of tea. The finer the tea, the more likely you are to extract everything in the first run. Take apart a cheap tea bag, and you'll notice that the leaves are powder or almost powder. A full leaf (you can buy any black tea from a reputable shop in a higher grade) will extract more slowly and give you a better flavour and probably yield to better second or third runs.