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I would say it could be a combination of harsh winter and a watering issue (I suspect drought). It look as though Idaho is experiencing a drought - has it been very prolonged? In addition, the joshua trees as a whole are actually succumbing to climate change:
http://www.desertsun.com/story/news/environment/2014/05/31/global-warming-joshua-tree-national-park/9729285/

If you are in fact experiencing drought in your region, I would water your joshua tree and observe it for improvement.

You may also want to inspect your tree for damage by the agave snout weevil (the only serious pest of joshua trees):
https://www.gardeningknowhow.com/ornamental/cacti-succulents/agave/agave-snout-weevil-control.htm

For more information on joshua trees, please visit the following link:
https://www.gardeningknowhow.com/ornamental/foliage/yucca/joshua-tree-information.htm

Joshua trees are very drought tolerant. If the soil is remaining moist too long you are probably overwatering. They need very well draining sandy soil. If you think about the Joshua trees in wild spaces they rely only on the rain they receive each year. This year we have had an excessive amount of rain in San Diego. I don't know if it is the same where you are. I would check the tip that is turning brownish yellow to see if it appears soft and rotting. If it is rotting you may want to remove the limb so the tissue rot doesn't spread to other areas of the plant. Cut well below the damage into healthy tissue. I don't know if it will grow a new tip. As with any pruning, clean your pruners/saw in a solution of bleach and water to remove any chance of spreading disease on your next cut/prune.

It can survive anything down to -23 celcius so as long as you can keep if from getting that cold then, yes, it could survive. Here is an article for more information: https://www.gardeningknowhow.com/ornamental/foliage/yucca/joshua-tree-information.htm

It does appear to be a young Joshua Tree.

Here is an article to help you with their care:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/joshua-tree-information.htm

This seems to be an older specimen. It will be normal for it to start falling apart as it ages. At this point, you can support the remaining trunks in any way that suits your. Some use ropes, some use boards, or any other methods.

Alternatively, if it becomes too much of a burden, you may need to remove the tree.

Here are some articles that will help:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/joshua-tree-information.htm

https://www.gardeningknowhow.com/garden-how-to/design/professional/when-to-call-tree-cutting-professionals.htm

https://www.gardeningknowhow.com/ornamental/trees/tgen/straighten-tree.htm

I suspect that it is overwatered. When you say there are rocks in the bottom of the container for drainage, are there also drainage holes? If not, I would get it into a pot with drainage holes. Also, adding a layer of rock is no longer recommended. Joshua trees do need more water when they are young, but as they mature, they need very little. While your tree is far from mature, I still would try cutting back on water, let the soil dry out between waterings. Also, the trees are susceptible to fungal issues and attack by insect pests. See this article for more information:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/joshua-tree-information.htm