It's actually Mitt kaffe (but Min kopp kaffe, since kopp is an "en" word). These are a bit tricky, sometimes it depends on an implied noun holding the beverage in question and sometimes I think it's down to the thing in question being uncountable.
We know it's mitt kaffe because the definitive form (the coffee) is kaffet (kaffen would be plural).
Edit: Oh, and "mina" is for plural things. Mina kaffen. :-)
Yes, when in neuter/ett.
It is a mass noun when used with ett (kaffet), so even if you have two packets of coffee beans or two cups of coffee, you will still say "mitt kaffe" (singular).
Used with "en", or a number, it means servings/cups of coffee. "Min kaffe" thus means my cup of coffee.
Since only the en-version can be plural, using "mina" will refer to servings/cups of coffe. You rarely say that though, instead it becomes the mass noun "mitt kaffe". (For "öl" however, that have the same en/ett use, it is more common to say "mina öl" for many servings.)
To specify f.ex. many packets of coffee beans, you say "mina paket kaffe", or "två paket kaffe". (Then it is "paket" that controls the gender.)
"kaffe" is neutrum (-et word) so it would be "mitt kaffe". But as Annika and Joakim explained above, "min kaffe" can also be correct, if you are not referring to the sort of coffee but the actual cup you are holding in your hands. It is the same with beer: "Mitt öl är Norrlands Guld/Mariestads/Falkenberg..." men "Det här är min öl, men du kan gärna hämta en egen från kylskåpet." "En" would refer to the bottle in this case.