Yes but no. The sun didn't form the Earth. The Earth and the sun both formed from a primordial star. You don't hear much talk about the star that actually birthed us, because theres very little we can say for certain about it. We can't even say for certain if Sol (the sun) is a second or third generation star because we only have vague notions about if the star that produced all the elements that aren't hydrogen in the solar system was long lived or went nova in under a billion years.

Probably

The molten core is hot because a whole lot of rocks crashed into each other. Also, nuclear decay.