Wouldn't it be more appropriate to say "a sheet of paper" instead of "a piece of paper"?

What does “never once” mean?

The expression "never once" means never ever, or never at all, or not even one time. It is used to emphasize that something should have happened, but never did. Ex: I moved next door ten years ago, and my neighbors have never once invited me over! Ex: In our 40 years of marriage, never once has my husband helped me cook.

What's "same old" mean? Does it mean old?

No, "same old" doesn't mean "old." It means the same ones or the same type of thing. Without change or difference. Ex: I've been wearing the same old things for a few years. I haven't felt like changing my style. Ex: We always talk about the same old things. Sometimes it's boring.

Does the word "off" has the same meaning as "right away"?

"Off" in this context is used as an adverb, meaning to be "away" or "at a distance" from the place in question. Thus, it does not have the same meaning as "right away," but can be used to state that something or something has moved away from its original location. Ex: Off we go to grandma's house! Ex: The criminal ran off into the distance.

I think maybe "Stop" can be used instead of "Enough" here. But I feel like there might be a slight difference in nuance. If so, how's it different?

There is a slight difference in nuance! "Enough" has the meaning of no more than what's already happened. One won't tolerate more of something. Whereas "stop" can be used in various circumstances, so it's not as specific to the situation as "enough" is. So the main difference is that "enough" expresses an amount of tolerance, whereas "stop" is just a command. Ex: Enough! I don't want to hear any more complaining. Ex: Stop complaining. It's annoying.

Is "suffering" skipped in the middle of "...for example, from..."?

Essentially yes, it is omitted from the sentence. Since "suffering" is placed in the beginning of the sentence, it is unnecessary to place it within the rest of the sentence. Removing "suffering" from the middle of the sentence helps keep it shorter and more concise. The sentence would be "...for example, suffering from insomnia or suffering from sleep apnea..." Omitting words that have already been mentioned is very common in English.