“I want/need/would like a coffee/a beer” is fine where I come from (UK) but we would not say “a tea/ a wine”. No idea why...
Because it is a noun that is uncountable so you say either do you want some tea when it is with no adjective or do you want black tea because there is an adjective
Well yes, but what causes the distinction from beer and especially from coffee? What you say is the result, but maybe there is no underlying cause at all, maybe it is just random that it ended up this way.
Yes, it is pretty much just random and changes from region to region. And even from occasion to occasion. If I were buying a cup of tea I would usually say 'Could I have a tea, please?' but I would never ask a friend 'Do you want a tea?' English is just weird like that, haha! (From Ireland btw)
It might be due to standardised measures for coffee (big, small, espresso, etc) and for beer (0,5 l or 0,3 l, sometimes 1 l pints) that everyone knows, when for the tea - the cups have different sizes, the jug, etc. and saying ”a tea” would be confusing because it is served differently in (almost) every place.
Is the "do" really necessary? The sentence makes sense in English without it.
I think wo would probably phrase it slighly differently in Czech. "Vy chcete černý čaj?" "Černý čaj chete?" But it might be acceptable with the right intonation, I will add it.