Such a fine hair you are splitting there... as a native English speaker, either case implies the state of "possessing" coffee or tea though when ordering we can also say "They are having champagne at that table." 'To have' in English is maybe similar to the French Prener such as "Je prend une cigarette" it implies not only taking and enjoying the cigarette, but also implies possession.
"So you either do the one thing either the other" <-- this does not sound like a proper sentence. Hence my inquiry about explaining to people estrange to the language what the "of....of...." part really means. I live with a Dutch guy who now says sentences like "or you do this or you do that", and it sounds rather confusing! :)
Your Dutch friend translates it incorrectly. It should be either...or. The sentence represents a choice between two alternatives. Imagine moving to a small apartment and you have to decide whether you put a closet or a table in the room, because only one item fits. You put either a closet or a table in the room. 'Je zet of een kast of een tafel in de kamer'. The example from your room mate probably suggest you have to make a choice between two activities because you cannot do both. You either do this or that, (but not both). Je doet of dit of dat, (maar niet beide).