Tywin left behind only 3 children. That seems like a very small family. How large was the average Medieval noble family? And why did he never remarry? Surely he’s not a sentimentalist?

Well, I’d direct you over to @joannalannister who would explain that Tywin was very much a sentimentalist when it came to his wife. 

On a more practical level, Tywin had dozens of nephews and nieces and cousins, so House Lannister wasn’t hurting for heirs.