I just bought a few games for the first time on GOG. I already payed for them, although I don't think the site has confirmed my payment yet. The shopping cart keeps filled with the things I already payed for and I can apparently still take them to checkout. Should I be worried? This is a confusing interface decision.

If the payment seems to have gone through on your bank/paypal/other payment system's end, definitely open a support ticket.
It could take a bit, since the weekend has just started and sales tend to bring some workload for the support team, but they'll clear the matter up.