What is this grammar rule?

I have here two sentences. I know both are grammatically incorrect, and I know of ways to make them correct. What I need to know is why they are wrong.

The first sentence: "For getting the degree, a person pay a lot of money." What grammar rule (I'm leaning on some sort of tense agreement) states that we must put an "s" after "pay" for the sentence to be correct? "For getting the degree, a person pays a lot of money."

The second sentence: "Furthermore, he or she studies major and takes tests each semester." The "studies major" part sounds awkward, but I don't know what rule of grammar says it's wrong. Obviously, it would sound more natural if it was written as "studies his or her major" or "studies for his or her major." But why?

Thank you in advance. English is my first language and yet, even though I know when a sentence sounds wrong, sometimes I can't say why. This is quite frustrating sometimes.