My understanding is there are three ways you can acquire a vault card.
You can use a Buy to purchase one, in which case you take a face-up one from the tableau and pay its cost to the bank.
You can gain/draft a vault card from the tableau, in which case you just take one irrespective of cost and pay nothing.
You can draw one from the top of the deck, in which case you take the top card of the deck, flip it face up so all can see, and pay nothing.
(Or 4: you could draft one of the three from the Vault location)
Are those all correct? And is a "buy" really the only time a vault card's cost matters?
You are correct.
Second question. Side A of the Student Store says you can use buys to spend 4 gold for a book, scroll, or research. Can you only take that option when you get a buy from the Student Store, or is that always an option no matter how you get the buy?
The 4 Gold for 1 Inteligence, 1 Wisdom or 1 Research is only available in the Student Store.