Need Iceland overnight layover advice

We are US citizens flying from Milan to US via KEF. Our overnight layover is 9.5 hours. Based on what i've read online, this airport is not streamlined for overnight layovers. So here's my plan, I just need some questions answered and flaws pointed out:

Land in KEF at 1:15 am.

Bags will be checked through to final destination.

Rent a car, leave the airport.

Sleep/nap in rental car. To get back on our normal timezone schedule, we don't plan to sleep for very long anyways.

Wake up, enjoy the sunrise around 5:30 from the coast nearby, grab breakfast, return car to airport by 8:30. Final flight departs at 10:40.

My question is, can we even rent a car that late at KEF? Is it safe/possible to sleep in the airport parking lot?

What security we will have to go through when we return to the airport?

Is this stupid?

Thanks in advance.