Yes, vent to ceiling. Leave tile as it is, it's perfect.

This picture shows tile beyond the hood BUT cabinets that do not go to ceiling...I think it looks wrong. 2nd picture shows paint past the hood, looks clean and nice.

I would extend the vent cover to the ceiling.

I don't think it's necessary to add tile above the vent hood.

The whole vent should go up to the ceiling, otherwise it looks as if they had run out of material or bought it cheap in an opportunity sale.

How did you end up finishing it? We have a similar situation but not sure how to deal with the crown moulding if the vent goes all the way to the ceiling.