How to deal with paying for dates:
(1) Split the first check, no matter what. If you fuck it up then you don't want either party paying for the whole thing.
(2) If there is a second check, and that second check comes at the end of a good date, pay for it and tell her to pay for next time. If it's not looking good, split the check and cut your losses.
(3) Third one is covered if you get that far.
(4) Wing it, you're dating now, you guys can figure something out, no need for hard-and-fast rules.
Addendums:
(1) If she really adheres to gender roles, that means you're paying the check. Deal with it, or don't go out with her, your pick, but starting any kind of serious squabble over paying dinner bills is a terrible way to get anywhere with anyone ever.
(2) If she talks a lot about being self-reliant or along those lines, you can stick to the above guidelines, but if she objects and wants to split a check where I said not to do so, then don't fight over it. It's just a dinner bill, and come on - this probably means she's offering to pay her half instead of having you pay the whole thing. If you're going to insist upon your right to pay more money for a date to the chagrin OF your date, you're an idiot.
(3) If you're in Japan and her father offers to pay, either (a) take the money or (b) meet up in secret if your pride is understandably slighted at the rather childish sight of you, a grown man, being handed money not by your own parents (which would be bad enough) but your DATE's parents to pay for the night. But don't raise a stink.
Basically, those guidelines should work, but if there's any extenuating circumstances with your date or her family that would create a conflict, go with whatever those circumstances dictate. The main thing isn't whether you pay the whole bill, it's whether you can solve the puzzle without making an ass out of yourself, and it's not a difficult thing to do.