I think, that it is not a problem with wrong URL settings.
It looks like page can not send AJAX request right, if the root of page is under another directory.
My page root is d.websupport.sk/www.trapez.sk/.
My checkout page is d.websupport.sk/www.trapez.sk/kontrola-objednavky
Everything works fine, but when AJAX request is sent, it is sent to d.websupport.sk/kontrola-objednavky instead of d.websupport.sk/www.trapez.sk/kontrola-objednavky.
Same problem is in cart page:
d.websupport.sk/www.trapez.sk/kosik
When I change delivery method, total price is not recalculated.
Because AJAX request is sent to d.websupport.sk/kosik instead of d.websupport.sk/www.trapez.sk/kosik