NP-hardness of existence of spanning tree with given maximal degree
Hint for part 1: What does it mean when $G$ has a spanning tree where every node has at most 2 neighbors?

Hint for part 2: Suppose you know that it is NP-hard to show that $G$ has a spanning tree where every node has at most 2 neighbors. Can you reduce this problem of at most 2 neighbors to the current problem of at most 5 neighbors?

This post is made out of user53923's comment as I have verified it is useful.