"Can I get one cup?"
In reality, placing the counter first actually emphasizes the number of items. You could say both nihon no biru o nonda, and biru o nihon nonda, with the first one emphasizing the fact that you drank two bottles, whereas the second one just means you drank beer - two bottles of it.
I answered the same and I'll say it is right. I base this on the lyrics for the opening song from YuYu Hakusho, where it says 二つ丸を, counter+noun+particle.
If someone could confirm it using more than anime theme songs, please?
カップ is cup, as in 'coffee-/teacup', and also for 'trophy' and 'bra size' kind of cups.
コップ is glass, though (like in English) it does not always have to be made of actual glass; ceramics and plastics may be referred to by this word as well.
Sidenote: グラス (from the Dutch "glas") is 'glass' - the material. So for wine/whisky glasses and such, you use that word.
Isn't the distinction between コップ and カップ that the former refers to handle-less drinking containers (tumblers/"glasses") and the latter refers to ones with handles (mugs/cups)?
I use an additional app called Simeji that allows me to switch to a Japanese keyboard.
If you download the keyboard swift key it gives you options for easily switching back and forth