What does ビー means in this sentence? I heard is 'soggy' or 'drunk/intoxicated', so that means something soggy from the nose... Is it a snot? And what is ?

Yes. I think it's a snot, nasal discharge.

means soggy and slimy. It's a kind of onomatopoeic expression.

is also an onomatopoeic expression, which means something appears in a rush and with a large content.

means "A lot of slimy snot is coming from the nose fast and furiously."

Hope this helps!

edit) There is another interpretation which is more likely:

"Hey, Mr.Snot, you should go out from the nose. Go!" It is said by a mother to her child.

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