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1. Then the story skips to her sixteenth birthday, during breakfast Alex announces that he is selling her cow, Libby, on Friday. (run-on sentence)

2. I prefer when a book is solely about a single character and Iron Bark Hill was entirely focused on Natalie. (run-on sentence)

3. The chapters about her past were in the right places, they provided more information about her perspective. (run-on sentence)

4. This novel seemed like it was edited by a professional, I could not find a single error. (run-on sentence)

Are any of these not run-on sentences, because I don't think some of them are but I don't know for sure?

I think if both, 3 and 4, had a semicolon instead of comma, they would have not been run-on sentences.

I would say the first sentence is a run-on and should be two different sentences.

In the second sentence there should be a comma after character, because those are two independent clauses.

Paragraphs 3 and 4 are not necessarily run-on sentences, but they would be more grammatically correct if they were two sentences or separated by a semicolon rather than a comma. They both contain two independent thoughts that are not connected by a coordinating conjunction.

I think you overused the comma, you used comas in places that you should use semicolon or period. That is why your sentences got tagged as run on. Try to use scribbens while profreading your reviews