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Ocotillo should be supported by larger rocks instead of being staked. "To help prevent a newly transplanted ocotillo from falling over or blowing down in a storm, large stones may be placed over the root area instead of staking, which often scars the stems. Leave two to four inches space around the trunk. Some degree of growth set-back is to be expected.", from Wikipedia.
In all the years I've grown cactus and succulents, there have been plenty of times a speciman has needed some type support. In fact, most of my colllection has at least two or three rocks to hold it in place. I like using rocks instead of stakes simply because they are more natural looking.

To the best of my knowledge, ocotillo is not the kind of plant that responds well to pruning. If you cut the branches, you lose the flowers, and end up with tangled and unattractive growth, losing the qualities you originally planted it for. That being said, I believe you could cut branches off at ground level, if they get too long. According to reports, these plants get 6 - 10' tall in the garden, so if this is going to be way to big for your needs, you may want to move the ocotillo before it gets much bigger, and replace it with something that will mature to the size you want, rather than trying to force the ocotillo to stay small. This article discusses ocotillo, you might find it useful: https://www.gardeningknowhow.com/ornamental/cacti-succulents/ocotillo/ocotillo-plant.htm

It is possible for termites to attack live plants, although not common. Treating the house won't affect the plant - according to what I've read you should circle the plant with a liquid non-repellent termiticide to control.
A more likely culprit is the ocotillo borer.

These pests are opportunistic, meaning they chiefly attack plants that are stressed to begin with. Here's an article that has some ideas toward having a healthier ocotillo. https://www.gardeningknowhow.com/ornamental/cacti-succulents/ocotillo/ocotillo-plant.htm
You can take some samples of your plant to the nearest extension service office; they'll be able to positively identify the problem, and give you control instructions. This link will help you find the nearest office:
https://www.gardeningknowhow.com/garden-how-to/info/what-is-extension-service.htm

Yes, they can easily be started from a cutting.

You will need to use a green cutting about 6 to 8" in length. Use rooting hormone on the end of the cutting to promote growth.
Place in a well draining pot with desert potting mix and water well every couple of weeks.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/ocotillo/ocotillo-plant.htm

http://aces.nmsu.edu/pes/lowwaterplants/ocotillo.html

Here is a link to refresh you on the growing conditions and care.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/ocotillo/ocotillo-plant.htm

I would increase watering.

Here is a link to refresh you on the care requirements.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/ocotillo/ocotillo-plant.htm