Or do guys pay all the dating expenses on a first date?
That depends on whether you want to see her again
We musn't forget that it's not always a guy and a girl on a date. I don't really like gender roles so I think it's just a competition of chivalry.
It depends on the people. I was taught that, as a gentleman, to always pay the bill. But many people split the check. Personally, I think the man should always offer to pay, but if the woman offers, allow her to pay as well.
Even in the same country, people have different opinions on it. As a girl, I think it's OK to pay for what you ordered. Nevertheless, I'd find it weird if a woman asked to pay for what the man ordered and the tradition with being a gentleman still stands in Romania. I think it would be impolite, though, for a woman expecting the man to pay most of the time. It should be all about respecting the each other.
Here in Australia, the man would probably offer to pay the bill but the woman would almost always insist on splitting it. If he pays the first time, she will probably pay the next time, if there is a next time. Alternatively, the man might pay the restaurant bill but the woman might by the movie tickets and coffee/drinks after the movie.