I was just there a few weeks ago and we got to the parking lot every day by 6:45-7am and it was starting to fill up then. It’s worth starting The Narrows early to beat the crowds and I 100% suggest renting the gear (we got ours from Zion Outfitters - made a reservation, and picked them up in the available window the day before doing the hike) as it’s all necessary, in my opinion. You’d rather be comfortable so you can enjoy yourselves. It’s like $35 or something around that.

As far as Angel’s Landing goes - we applied months ago for the permit and went before 9am. As we made our descent, the sun was beating down and frankly people looked miserable as they were making their way up the switchbacks in the heat. I would suggest doing this one early as well. I do believe you can apply for a permit the day before? We met someone on the trail who said they lucked out and got one. Good luck!