Zimone and Dina doesn’t need to have been under your control when the first card is drawn for its ability to trigger. As long as you control it when you draw your second card in a turn, that ability will trigger.

For the activated ability, repeating the process means the entire effect, but not the cost. You don’t have to (and, in fact, can’t) sacrifice another creature. You’ll draw another card, then you may put another land card from your hand onto the battlefield tapped. As this is your second time through the process, the ability will finish resolving.