It's perfectly normal in Danish. You'll hear Danes say "Hun tager til lægen" just as often (or more often) as "Hun går til lægen." It may sound strange to non-natives, but Danes find nothing strange about it. You probably won't hear us say "Hun tager/går til doktoren" quite as often, however.
Why is "takes herself" not accepted? As in the English formal register eg "She should take herself to the doctor". Or would that have to be "tager sig"?
No, the Danish word "gå" is used both like the English "walk" and "go", and possibly a few other ones, depending on how you use it.
"Jeg går på vejen" means "I walk on the road", but "Jeg går i skole" means "I go to school" even if your parents drive you to school every day and you never actually walk. However, "Jeg går TIL skole (hver dag)" means "I walk to school (every day)," and yet "Jeg går til fodbold" means you play football in a club or something, not that you walk to see or play in a game.
So it's not as simple.
Det ville i hvert fald blive en lang tur hen over Sibirien og ned gennem USA :) Men i det her tilfælde er det ikke så meget et bestemt sted men en person du går til. "Jeg går til lægen" kunne jeg da godt finde på at sige uanset om jeg så gik, kørte eller tog bussen.
You can. I think "lægen" is more common here, though. I believe "doktor" is mostly used as a title before a name.
How do you know when to use tager, and when to use går, or would either one be acceptable?
"Tage til" is usually used if you have a certain goal you're going to. "Gå til" can also be used, but only if you can reasonably assume that there's walking involved. "Jeg går til lægen med bilen" sounds odd, you'd use køre here instead.
Betyder "læge" "physician" på engelsk? Selvfølligt betyder "doktor" "doctor".
Læge is the profession, doktor is the title. Calling a physician doktor is uncommon, mostly children's speech.