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It's not a requirement. The reason many vipassana guides say to breath through your nose is because most people can easily feel the air at the tip of nose and it gives them a clear point of focus. You can breath through your mouth too. Some people prefer focusing on air coming into the stomach instead.

When I have a cold or some other build-up in my nose, I just breath through the mouth, no problem.

Nope, don't worry about it :)

Don't get hung up on the minor detail. It's super easy to do with meditation but unless you are following a very specific style, it's just not important.

It doesn't have to be the nose. You can use the rise and fall of your stomach as your focus. It is merely a point that allows you to notice when you've become distracted.