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Do flights from Switzerland to Germany have an immigration check?
No immigration check (i.e. you are very unlikely to see a police officer) but airlines frequently do check passports. Even if that's not always the case, I wouldn't rely on not having to show your passport. There is also always a chance that a customs officer wants to talk to you.
Is my Legitimation Card enough for traveling between these two countries?
Probably not but I am not sure. Even if it shouldn't be accepted (and I don't know that for a fact), it could still be enough to bluff your way through. I once flew with someone who was able to board a flight out of Switzerland with a residence permit from some other European country (after losing their national ID card shortly before boarding). But it was a flight to her country of residence and it required a whole lot of discussion.
Does the US consulate provide a letter stating that they are with my passport while issuing the visa or something?
I don't think that would really help you. Airlines are more interested in checking that your document is on their list and reducing liability than understanding your situation.
Note that it is also required to hold a passport while in Germany and you probably have to worry about the flight back to Switzerland.