I understand 'min' is for 'en'-words and 'mitt' is for ett words. My understanding was 'kaffe' is an 'en'-word if you mean you want a cup of coffee, but if you want to refer to a certain kind of coffee it would be handled as an 'ett' word. Does 'mitt kaffe' then mean the person is saying something along the lines of 'my [brand / type of ] coffee?
There are some noun endings you could look for in the chapter on nouns which give clues to gender. http://www.readersstuffz.com/downloads/ebooks/Language%20Books/Swedish/Swedish%20-%20Essential%20Grammar.pdf
En kaffe sounds just about acceptable to me if used for ordering. I'd personally always specify the unit, i.e. "En kopp kaffe, tack". Of course I'd never order "ett kaffe, tack" either.
Min kaffe sounds abominable though. "Jag hämtar min kaffe" does not sound natural at all to me. I would always refer to it as "mitt kaffe". Is this a regional thing I've somehow managed to avoid for 34 years?
is the mitt/min difference just boil down to en ett difference s? I see cognates with Swedish min and English mine (archaic) as well as Swedish din and English thine (archaic). makes me want to research premodern and Old English before the loss of most gendered nouns to find more parallels.
Spelling doesn't always make sense, but here it actually does, since the i sound in mitt is short, the word is pronounced as mitt, not as mit.
A better question would be why min is not written minn, since that is how it is pronounced, but for that one I don't have a good answer. It's just down to convention, I think.
A bisarre little quick of Swedish: coffee is an uncountable mass noun that takes -t, so the coffee is "kaffet" and my coffee is "mitt kaffe". BUT you CAN hear people order "en kaffe" in a café because sometimes with t-ending mass nouns in Swedish, if you want to talking about one specific unit of that thing, it switches genders. Another example is mejl, which is absolutely a t-word. The e-mail is "mejlet". But you sometimes hear people asking for each other's e-mail addresses my saying "Kan jag få din mejl."
Swedish Radio's "Språket" program did an interesting segment on this, but no need to worry about it at this stage. Focus on kaffe being a t-word. Kaffet. Mitt kaffe. :-)