Could a High-Altitude Nuclear Explosion cause _____ to happen?

No problem. As fast as you destroy it, the Sun makes more. No big challenge.

Drop dirty snowballs every day, and wait months for the water vapor to get over the target continent. And hope it did not diffuse globally.

It isn't going to be one day's worth of UV-B that will do us in.

No. Even it it could cause a hole in the ozone layer (not that it would) that would not destroy the land under the hole. UV light that would be allowed through by the hole is harmful to the eyes and skin but short term exposure would not cause significant problems. Plant life would actually thrive as a result of the additional UV.

Your worry is so 1940's old. Please research your subject so you can ask more informed questions.

This was a concern of scientists working on the Manhattan Project in 1945. As you might have figured out by now, nothing of the sort happened then, or at Hiroshima, or Nagasaki, or all the Bikini Atoll tests in the fifties. So, why would you think the same could happen now?

The atmosphere is not a piece of fabric. It does not 'tear'. You are applying metaphors improperly.

Blowing a hole in the atmosphere with a bomb is like digging a hole in liquid water - it can't be done.

This was actually done...they wanted to blast a hole through the Van Allen belts so that they could go to the moon. The blast made the problem worse.

Because this doesn't sound suspicious at all