Not even remotely possible
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The route specified on your link is 2,000 miles long and if you drove 10 hours per day, it would still require four days on the road. You've only got 2-3 days of travel identified and the most direct route (I-70 to I-15) is 1080 miles and in the real world, this will require 19 hours of travel when you add in fuel stops, food, lodging, traffic slow downs. Traveling at or slightly above the speed limit, will require two days of 10 hours of travel.
You might be able to squeeze in Death Valley, on the way to San Diego -- but you need to expect to be on the road much of 12 hours each day if you're going to squeeze in anything off the Interstate Highways.
One of my favorite routes in the west is I-70 through Utah -- here is a field report.
And here are some tips for conducting a Speed Run!
Mark
Not just time stopping you.
It is not only time which will prevent you from driving your planned route. Snow and weather will as well. The Trail Ridge Road over the Rockies will not be open in April, and is not guaranteed open at any time of the year. It was in August one year, when it was closed by a blizzard, and again when I wanted to drive it, in September a few years later.
And when you do drive it, the speed is strictly limited. You can't possibly drive at full speed across that road. And why would you want to, even if you could. That magnificent and spectacular landscape and scenery at the top of the Rockes is to be savoured.
Best you follow Mark's advice, and leave your sightseeing till you have the time to really enjoy this beautiful corner on the globe.
Lifey
A case of searching for deals.
For Joshua tree you want to cut through the park from Twenty Nine Palms and then pick up the route through Palm Desert as described by Buck.
As for car rentals you need to search around and see what deals are about. They all offer much the same, but with different offers around all the time, even form office to office. You can check the RTA reservations service towards the top right hand side of the page.