The non-ceremonial stuff is PR and goodwill for the UK. She's essentially fostered economic cooperation between the Commonwealth for decades. That has been quite important and should not be understated. The American view that her duties are fluff is a lazy take.

Can she set her own agenda, or does she essentially work for the elected government?

Sets her own agenda, but I mean she holds weekly meetings with prime ministers. It's all closely coordinated.

I saw that Prince George is third in line. I wonder how that would work if he ended up a child king? In that case, I’m guessing he would just be doing what he’s told to do.

Regular meetings with the prime minister must count as having some power.

Technically, I believe she can dissolve parliament in NZ and Australia.