날리다 is 'make something fly' — the causative of 날다. In these lines people fling attacks and slaps (소심하게 날려본 뺨따귀 'a timid slap to the cheek'), send finger hearts (하트를 그렇게 날렸는데), launch paper planes, and blow things away (시원하게 날려드리도록 하겠습니다). -게 supplies the manner. 날려보내다 adds 'send off, let go' — 애들을 키웠으면 이렇게 날려보내라 'once you've raised your kids, let them fly'.