While it's good to learn "doktor," I think it's far more common to hear and see "læge," especially on signs, like "dyrlæge" or "tandlæge." The hospital's website (Bispebjerg) only uses "læge." That might be more useful to know, especially during an emergency. :)
"Doktor" does not necessarily mean "læge", but a person with a higher academic degree. http://www.denstoredanske.dk/Erhverv,_karriere_og_ledelse/P%C3%A6dagogik_og_uddannelse/Universiteternes_eksaminer_og_akademiske_grader/doktor
Doktoren is a bit old-fashioned. Our old family doctor was called dr. Floding by all his patients. It is a bit more respectful than læge, and since noone cares about hierarchies in DK anymore, it is a dying expression :)
Actually Doctor comes from Latin, as the dictionary says: http://ordnet.dk/ddo/ordbog?query=doktor I suspect the origin would be more via neighbor German than via English , I have realised how wonderful amount of words (specially academical ones) comes via German, and Doktor is identical there. But anyway, this word is avaliable in almost every European language I know.
I would think that in a lot of cases where words in english and scandinavian sound alike, the english is derived from the scandinavian. :-) The vikings, you know.
I'm also taking Danish lessons in Denmark and they only use "læge". Using "doktor" is not correct.