Where I live (Europe) it is also common to wait until the other person arrives. Usually, you aorder your own drink, but that's it. If you order for someone, like deciding what they should drink, you pay. If you don't want to pay for the other person, you let them choose what they want to order themselves.

He ordered things for her, not letting her choose what does she wants and pays behind her back, it is his doing. He should take responsibility for his choice.

I usually pay half the check. But I would also been annoyed that he ordered for me. That would have been the end for me. It is not OK since you don't know posible alergies, likes or dislikes. Neither her budget.