The game is locked when the ML World Series ends, so any moves you want to make must be done before the end of the WS. Rollover happens approximately three days after the WS ends.
As for resigning potential departing FA's, the best bet is to wait until the regular season ends. You can sign them to extensions before then if you wish, but there's always a slight chance that they could suffer a major injury in the last couple of games, and you may want to reconsider whether you really want them to return. So to play it safe, wait until the regular season has concluded. If you make the playoffs, then wait until you're knocked out. If you make the WS, then do it late in the Series but before it has concluded.
Minor league promotions can be done at any time. If you have teams in the minor league playoffs, then you may want to wait until they are over. Unless you have good prospects who are due to be promoted to a level that's still alive in the playoffs. In that case, you want to promote them now so that they get more playing time.