What chemical processes occur in tea that spoil it after some time?
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.