Did the NSDAP help to fund Lufthansa?
NSDAP was the elected ruling party in Germany from 1933 to 1945 (only the first part of it elected though); thus Lufthansa was given government money, not "Nazi"-money. During this time, they made the Swastika the flag of Germany. American Airlines sport the American flag all over their tail, and I'm sure they (at some point) have received government money as well. If you replace every occurrence of "Nazis" in your question (including title) with "government", you'll notice it becomes slightly less provocative, and the parallels to other airlines and other countries become readily apparent.

When the war started, I would assume the military requisitioned all the assets they needed including airplanes and services from Lufthansa. I'm not much of an historian though, so I can't give you any details, but I'm sure they're out there to find. I realize this doesn't really answer your question (Lufthansa's involvement with the German government), the point I'm trying to make is that it's very hard to answer, as with any airline and any government.