Any examples of Craft Commerce checkout process for PayPal or any other gateway?
Craft Commerce ships with a templates/shop folder that has a working check-out processes using the Dummy gateway adapter and some default products that get populated during installation.

After installing it, you'd access it by going to yoursite.com/shop.  You can use a test credit card number like 4222222222222 to complete checkout and convert a cart to a completed order.