Can a unit be played onto an occupied battlefield and also move other units there in the same action?

No, these are separate actions. Arachnoid Horror's ability only lets it be played onto an occupied battlefield — units already in your base have already been played, so its text does not let you also move friendly units there in the same action. The same applies to Rengar, Trophy Hunter: playing him and moving units are different game actions, and moving a unit requires there to be no chain and no showdown, done on your turn when you could otherwise take any game action. You can move units to a battlefield and then play Rengar there afterward, but you cannot do both as a single action.