Does brewing tea for longer result in higher caffeine content

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.

A rainbowy layer is almost certainly oil. (This is called thin-film diffraction.) Earl Grey uses oil of bergamot for flavoring, and likely cheap teas you've used have oil-based flavorings as well. It's not surprising that the oil eventually separates, and there's not really any way to avoid that without significantly modifying the tea, or using tea that doesn't have quite so much oil.

I would definitely expect the flavor to be stale after it's left for a while - aromatic compounds are by nature volatile, and they'll slowly escape. You might be able to prevent some loss of flavor by putting the tea in something airtight (preferably a completely full container, so there's not even air on the surface) and chilling it, but it's going to be a losing battle. You said the color was stale as well; I don't know exactly what processes would cause that, but in general, the only way you're going to be able to slow down any chemical processes is by chilling the tea.

In the end, the real answer is simply that it's best to drink fresh tea. Your time would probably be better spent getting to where you can make it quickly - for example, get an electric kettle that can rapidly boil a single cup worth of water.