Hi,
1. Check that the "auto select shipping and payment methods" option in your HikaShop configuration is turned on. If not, then you found why you get that behavior 
2. It really depends on how you configured your checkout workflow.
If you have only the payment view on the current step, the system will transfer you to the next page when you select the payment method. But if you have another view which can be influenced by the payment method selection (like the cart where the payment fee needs to be recalculated after changing the payment method) on the same step, then the system keeps you on the same step so that you can check the changes before going to the next step.