Depends there is a difference between you to going out amd him asking to take you out. Example he asks "we should go mini golfing" if says soemthing like were he isn't directly saying let me take you out in a date or he isn't implying that then it's the cas eof where you each pay your own share. If says something like "hey there is a great Italian place that opened, would you like to come with me to try it?" That's the type of question where he is asking you to occompany him so there for he should pay cause he is asking you out to that place.
I'm surprised at all the splits lol. I would definitely pay. To me, not paying for something so insignificant even if it is just a continuation of traditional values seems douchey or at the very least it shows a lack of interest in the woman sitting across fro you.
Split it - they split it! I wouldn't let someone pay for me especially if I couldn't guarantee I'd like them. Now once you've been with someone awhile and they wanna take you out on them that'd be a nice gesture.