The answer to shticky guy’s points is that we take the lulav and esrog for the rest of the days medirabanan. Medioraysah we only take it one day. Shaking the lulav and esrog on shemini atzeres is a gezeirah legezeirah. So we don’t do it.
We don’t make a bracha on the sukkah because we also do things as shemini atzeres and it would be tarti desasri to make a bracha.