I don't believe "than he" is technically correct, it should be followed by a verb - so "than he does"
Actually, "than he" without the verb is technically correct, but not natural. It's more natural and far commoner to include the verb after the pronoun.
You are correct. "than he" is correct, "than him" is not correct. This is a common misconception in English. https://www.grammar-monster.com/lessons/than_I_me_than_he_him.htm
Nope, you need something to establish comparison between sister's drawings and his, so you need either conjunction „niż” or preposition „od” here.
That said, „Jego siostra rysuje gorzej od niego” should be accepted.