Been blind in my left eye since birth and I've always had balance issues. I don't think it's just a depth perception thing, because I've had 40+ years to adjust to that. I believe it's neurologically more complicated than that.

But yes, doing exercises to improve your balance will help to a degree - you'll have better/more refined muscle coordination to help you compensate and recover from a loss of balance more easily.

Definitely is playing a role in your ability to perceive depth. As you mentioned aging affects balance as well. The vestibular and proprioceptive systems work off the information the eyes and brain give them, helping with stability and position whether standing still or in motion. It takes the vestibular, vision, and proprioceptive systems working together to keep things level; but it is the vision which provides most of the necessary information. Try to balance with your eyes closed (be safe) and notice how difficult it becomes. Vision and balance are closely related.

There are some people who when they close their eyes they lose their balance because they have no sense of their body. Most people however have a sense of their body and it's positioning. So, you might want to practice the moves in a safe environment with your eyes closed just to get a sense for how your body feels. Then when you do it while seeing, try to use your body's sense as your primary sense, and your vision as a secondary guide.

Yes, very much so. I've been limited to one eye (and not my dominant one) for the last 4 years, and my balance still hasn't worked out all the kinks.