is there a connection between Gemini and ingeminating?

Yes, geminus 'born at the same time, twin-born', gemino 'to double', and ingemino 'to redouble' are all related. The verbs are denominal, and the noun is no doubt related to gigno 'to beget, bear, bring forth', nascor 'to be born, begotten', genus 'knee; birth, descent, origin, stock, child', etc.