Enter and exit through Joshua Tree that will be fastest. By 2:30 - 3pm don't bother with the shuttle although you will probably have a wait to get in but it shouldn't be more than 20 minutes. Sounds like you are staying overnight in Palm Springs? Or are you staying in Joshua Tree? Makes much more sense to stay in Joshua Tree or adjacent Yucca Valley (many more hotels) than to drive back to Palm Springs.
Here a link to a map of Joshua Tree National Park so you can identify locations. There is one main road (Park Blvd) with a couple off shoots.
https://www.nps.gov/hfc/carto/PDF/JOTRmap1.pdf
You only have until noon on Monday? Not sure I'd call that a "half day" since sunset occurs about 7:15pm. The earlier you enter the park the less wait, the easier to park.
There are a lot of one day itineraries here on the forum that could help you.
But in an attempt to answer your questions of which there were many lol -
Yes it is safe to drive around the park at night, no problem.
Cap Rock has a short hike and plenty of parking with picnic tables if you are looking for a pleasant place for night sky viewing - although that can really be done almost anywhere. Cap rock at least has safe parking and picnic tables you could even lie down on.
Most of the hikes in Joshua Tree ARE short - just a mile, some 1.5 miles round trip. Cap Rock that I mentioned is short but not much to see except the giant cap rock that you can photograph from the parking lot.
Sunset at Jumbo Rocks is great. You park at the entrance to the Jumbo Rocks campground, not many parking spots there, if there are none then you can park along the street too or enter the campground where there are more parking spots throughout in various spots near restrooms, etc.. From the front/main small parking area, it's just a 5 minute walk where you can be surrounded by giant boulders and awesome rock formations - giant round boulders on top of giant flat boulders, etc..
If you like night photography you could consider this short evening trail. Bring a flashlight if there is not a full moon.
https://www.tripadvisor.com/ShowTopic-g143037-i4367-k11376318-Full_moon_hike-Joshua_Tree_National_Park_California.html#90430599
Here's a post with my answer from someone who only had half a day for the park:
https://www.tripadvisor.com/ShowTopic-g143037-i4367-k11254642-A_few_hours_in_JTNP_advice_please-Joshua_Tree_National_Park_California.html#89514004
Here's a post regarding someone who had one day - you could split this up:
https://www.tripadvisor.com/ShowTopic-g143037-i4367-k11149063-One_day_trip_from_LA-Joshua_Tree_National_Park_California.html#88650322
More info - disregard the details about hiking guides as that doesn't apply to you but there are lots of thoughts about night photography and hikes that you could squeeze in if desired:
https://www.tripadvisor.com/ShowTopic-g143037-i4367-k11035826-Looking_for_hiking_guide_and_photography_locations-Joshua_Tree_National_Park_California.html#87
As you exit the park to head to OC, the best vegan place right there in downtown Joshua Tree is the Natural Sisters Cafe
http://www.naturalsisterscafe.com/
Edited: 5 years ago