Does the phrase “clouds gather into a storm” make sense?

The phrase “clouds gather into a storm” would make better sense with the word "rain": “clouds gather into a rainstorm”.

Yes, especially if you're talking about a person getting very annoyed ("his eyes gathered into rain clouds"), but also about a literal weather incident.

Yes. That's a good way of say it to. Many small bad things come together to form something serious.

Yes.

Consider the following excerpts from published literary works:

"But thoughts of Mom and May were dark clouds gathering into a storm in his head." - The Rock Star's Scheming Bride by Emma St. Clair

"When he looks up he realises it's not the sun setting but rather the clouds gathering into a storm." - Ocean Eyes by Stephen Hobbes

"He also noted higher up darkening cumulous clouds gathering into a storm-like brew." - Thundercloud: The Oddities of Young Man's Journey into Adulthood by Ronald Joseph Kule