Should dricker work her as well as dryck?
No. "Dricker" is a verb and is not something you could have. An example:
but English "to have a drink" means 'to drink something', I'm guessing "I have a drink." in this context means "I have a drink [in my hand]", right?
That is correct.
Very good point. I was taking it the other way this whole time. Whoops! Thanks!
What about "Jag dricker en dryck"? While it's not preciselly I 'have' (own, possess), but to "have a drink" also means to drink a drink :)
That is correct. It sounds overly specific though but yes, it is correct.
Why "en" and not "ett"?
Most words in Swedish use "en" (about 75%). You really just have to memorize which use "en" and which use "ett".