I have lived in the US all my life, and I have never heard anyone say that they have an errand at the doctor!! We would say, "I have a doctor's appointment" or "I have an appointment with the doctor". If for some strange reason there would be another point in going there, and I can't imagine what it would be (maybe to finish the magazine article that you started in the waiting room?!!), we would never call that an errand. We might say, "I need to stop at (or run by) the doctor's office to (or for)..."
Sorry if I confused you. You are 100% right - you will have heard that phrase. And you might also hear phrases like "I'm going to my friend's". In a way, it's shorthand for "I'm going to my friend's house". So there are words that we don't actually say/write, but we just assume that the listener/reader understands them from the context. Hopefully I haven't made that more confusing!
And hopefully Simon's clarification on the Norwegian helps more than I did.
I wonder if that might require something like "Jeg har et ærend hos legens" (i.e. with the "s" right at the end to denote a possessive); as that is really that you're at the doctor's rooms / doctor's surgery / etc. in my mind. As I'm not a native Norwegian speaker, I can't be sure on that bit, but I think I'm right on the English bit.