Is this normal?

[m_c]

Woohoo, I can finally reply.

I wouldn't be too worried about any of that.
The pulley runout isn't ideal, but as long as the belt isn't going tight at any point, I wouldn't worry.

Runout on three jaw chucks is perfectly normal, as chuck scrolls are not a precision item.
If you need better tolerances, then you either need to use custom bored soft jaws, a collet chuck, or a 4 jaw chuck.

[A_Camera]

If you need better tolerances, then you either need to use custom bored soft jaws, a collet chuck, or a 4 jaw chuck.

You are right, the pulley play does not matter that much, unless the play is transferred to the chuck.

This is a 4-jaw chuck, it is a self centring type, I don't know if that's good or bad, I think in some situations it is good, in some it is bad.