Plant is turning real light color, almost white.
Too much shade can actually cause the leaves to become lighter.
Increase sunlight to your plant.
Here is a link to refresh you on the care requirements.
https://www.gardeningknowhow.com/houseplants/ponytail-palm/ponytail-palm-care.htm
What is the root system like? Does it travel or root ball?
Considered a desert dweller plant, they have a fairly shallow root system.
In a container the root system stays compact but can even break pots.
In the ground the roots may spread out to seek water.
During the winter, my cats have somehow jumped into the palm and have eaten the leaves or have destroyed the leaves by getting into the palm. It now looks horrid. What can I do to make it look better without killing it or inviting disease?
I would take some good sharp scissors or yard snips and trim the damaged parts of the leaves to help preserve the appearance of the plant. One thing you will not want to do is prune any woody material as that would leave an open trunk with no greenery and make it extremely vulnerable to mold and mildew. Other than that my only recommendation is to keep following the care plant for a ponytail palm (see links below) and to try to keep it away from your feline friends.
https://www.gardeningknowhow.com/houseplants/ponytail-palm/ponytail-palm-care.htm
https://www.gardeningknowhow.com/houseplants/ponytail-palm/pruning-ponytail-palms.htm
Can I take the new plants growing from the trunk and base and grow them? If so, how?
You sure can! This article will tell you how: https://www.gardeningknowhow.com/houseplants/ponytail-palm/propagating-ponytail-palm-pups.htm
Ponytail palm seems to be sick. Bark his come loose from trunk. It has a new start or baby on the side. What can I do to save plant? Or can I cut new start and start a new plant?
How is the rest of the plant? Wilting, yellowing or brown leaves?
This could be a fungus or even root rot.
Make sure your watering correctly and that the soil is not soggy.
I would isolate the plant and monitor the situation.
Yes, you can propagate the 'pup' from your plant.
Here are some articles to help you and refresh you on the care requirements.
https://www.gardeningknowhow.com/houseplants/ponytail-palm/ponytail-palm-care.htm
https://www.gardeningknowhow.com/houseplants/ponytail-palm/ponytail-palm-care.htm
I've had the plant for 4 yrs and now I have new shoots sprouting. I'd like to know if there is a way to remove them without damaging the trunk, to start new plants
Absolutely! Make sure they're at least 4" tall before removing them. Typically the best time to do this is in the spring, but if the pups (new shoots) are big enough and you keep them away from severe heat, you might be able to pull it off.
This article will tell you how:
https://www.gardeningknowhow.com/houseplants/ponytail-palm/propagating-ponytail-palm-pups.htm
I have a very large ponytail beside a fence line, which is pushing the garden fence over. I am wondering can I cut the base, approximately 6 to 8 inches, and will this cause stress to the plant? Will the scar cover over or do I need to re-position the plant away from the fence line. The ponytail has been in the position for approximately 15 years. Thanks Steve K Bribie Island QLD
No, cutting into the bulb or base of your Ponytail Palm would most likely lead to the death of your plant.
Here are some articles with more information.
https://www.gardeningknowhow.com/houseplants/ponytail-palm/how-to-move-ponytail-palms.htm
https://www.gardeningknowhow.com/houseplants/ponytail-palm/pruning-ponytail-palms.htm