LA to Las Vegas in seven days

[sidewalkballet]

Hello! I'm new to this forum and have looked through lots of previous posts - tons of great info here and I really appreciate it all.

We are two people traveling from LA to Las Vegas in July. We have booked a campervan (edit: a retrofitted minivan, not an RV) for seven days and seeking advice on places to see and general roadtrip/campervan tips.

We're planning on speeding through California due to the time constraints and the heat. From our research so far these are some spots we're looking to hit:
* Jerome, AZ
* Sedona, AZ
* Grand Canyon
* Lake Powell/Page, AZ
* Bryce Canyon, UT (?)
* Zion National Park, UT
* Hoover Dam

Seeking advice on how many nights to stay in each spot, and if there's anything in this area we shouldn't miss, we'd love to hear it. We're looking to do smaller hikes (less than 4 hrs), big fan of watering holes and waterfalls, scenic routes, good food. We're from DC and really excited to explore this part of the country.

Also, do we need to book a campsite for each night, or can we get away with some nights being loose / parking on the side of the road, etc?

Thanks in advance!