Learn Swedish: Lars dricker Karins kaffe. - Lars is drinking Karin's coffee.

In Swedish, the possessive 's' is added directly to the end of the name or noun without an apostrophe. So the English "Karin's" simply becomes Karins.

Lars already ends in an 's'. How would we say "Lars's coffee"?

If a name already ends in 's', 'x', or 'z', Swedish doesn't add anything at all! You don't add an extra 's' or an apostrophe. "Lars's coffee" would simply be Lars kaffe.

Since it is her specific coffee, why don't we use the definite form and say Karins kaffet?

Whenever you use a possessive word (like Karins, or words like min for "my"), the noun that follows must be in its basic, indefinite form. So it is always Karins kaffe.

Does dricker translate to "drinks" or "is drinking"?

It can be both! Swedish does not have a separate continuous tense (the "is doing" form). Depending on what makes sense in English, dricker means either "drinks" or "is drinking".