No, he didn't. He offered to cover a case of beer for after our game at the park, but he didn't have a car, so he asked me to bring it and he'll pay for it. Afterwards he came to me (I never brought up the topic of money) and asked for my Paypal details. How am I in the wrong here?

The whole "I pay this time you pay next time" is already covered by "everybody brings a case once in a while", but if the guys without cars just ask someone else to pay for it when it's their turn it won't add up.

If you're going to make assumptions, perhaps start with those that don't paint me in a bad light.