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.
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.