I am interested in two cases: 1) the congregation is davening mincha in a rush, and doing a "hechi kedusha" (btw, what is the etymology of this phrase)
2) an individual arrives late for mincha and is able to start shemona esrei with the chazzan.
In each case, should one wait until after kedusha to begin shemona esrei, or try to start with the chazzan and go along word-for-word. In the latter case, should the individual say the entire kedusha or just the standard communal responses? How should he conclude the beracha?