Yes : ) it "comes" from us : ) In Polish we have Niemcy which preetty similar sounds like you write it : D
As a German, I can confirm that the root meaning of the word is "mute" because… apparently our medevial ancestors didn't speak Slavic all that fluently. What can I say...?
Yep, the German was the one who "nie mówi / nemluví / не молвит (doesn't speak), which gives the adjective niemy/němý/ немой (mute) and hence, the substantive meaning "the mute one, the German : niemiec / němec / немец.
I used Polish, Czech and Russian but it's used in most Slavic languages (not sure for Bulgarian but for the rest, it is) and it was also borrowed by non-Slavs like Romanians or Hungarians (they say német).