Randy Newman is one of my all time favourites! I love a lot of his songs and albums. His latest "Harps and Angels" is a great one!

He has hos own rolling style of playing the piano, but I don't think there are any agreeable official terms for his particular playing of that song. If I were to describe it, I'd say he plays a mix of ragtime or New Orleans style piano (because of the rolling technique), and Broadway style (very little use of the sustain pedal, and progressions we know from jazz standards with a lot of embellishments).

"You've got a friend in me" is a film piece, and his Broadway sounding piano suits that genre like paintings on a wall. I think the fact that Newman is able to combine pop and broadway is part of why he's so sought after in Hollywood..