Yo,
Wikipedia suggests that the window cleaner contains ammonia, alcohol, dyes and fragrances diluted in water. Bleaching agents often contain sodium hypochlorite diluted in water. Wikipedia indicates that Comet contains sodium dichloro-s-triazinetrione dihydrate.
Avoid mixing bleaching agents with ammonia based cleaning agents. (this from a controls engineer not a qualified chem type dude)
The coffee and tea stains add character to the cup. A bit of detergent can be a good thing. When I want to rid a cup of the discoloration I use pure liquid bleach that eliminates the stain in seconds then rinse the cup - lots.
Anecdotal: my former colleagues at the thick film hybrids plant I worked at many years ago cleaned my heavily stained coffe cup in the ultrasonic cleaner normally used to remove soldering flux. The QA guys would have had a seizure if they had known. It came out spotless. The cleaner was a material called Prozone. The data is somewhere on the Loctite Henkel website, but it is being a pain in the ass tonight and won't let me search.
"Avoid mixing bleaching agents with ammonia based cleaning agents. "
E.g. don't spray ammonia/Windex on a cloth, then sprinkle on some Comet cleanser or spray a bleach-containing spray solution onto same.
The combination of ammonia and hypochlorite will form a dilute solution of hydrazine (yes, the rocket fuel), which is very toxic (causes liver damage). But, it's also a heck of a good solvent (will dissolve/destroy most any organic coating save fluorinated compounds), and works well to take creosote stains off the inside of a windowed woodstove door. The standard instruction "use in a well-ventilated area" would apply here.