I have only read good things about drinking green tea. However, to get the benefits out of it, you need to drink a lot of it. Or so I have read. I drink it daily but probably not enough for it to be of great benefit but it keeps me from drinking the bad stuff (I.E. soda).
The stains in the mug is proof that you are seasoning the cup. That is, each cup will taste better and better. :-)
You mentioned it getting cold... My method is to use a tea ball with hot water to make the first cup and I place the hot, soaked used tea in another cup until it cools. Water inside the tea ball drips out. Then I fill that cup with cold water. That is how I get my second cup of green tea in the morning. Since I use loose leaf tea, I mix the green tea with stuff called Luscious Licorice. Makes the green tea even better!
Drink on!