Checkout is not advancing to PayPal screen when checkout for gTranslatePro

Hi,

I checked your checkout but I don't see the same problem as you.
Regardless of the language (I tried all of them), after I login with the "Support" account you had provided before, I don't see the address area appear. And if I refresh the whole page, I see again the login/registration forms (where I shouldn't see them).
That explains why you would be redirected back to checkout when clicking on the finish button after the login/registration since the checkout doesn't consider you logged in when you press finish.
This indicates that the fact that the user is logged in is lost after the end of the page processing.
This information is stored in the user session. so that means that you have an issue with the user session not being kept from page to page or that it's kept but not the information of the user stored in it by HikaShop.
So first, make sure that the user synchronization plugin is enabled with the access public in the Joomla plugins manager. It's this plugins which manages the session for HikaShop so if it's not there, the login / registration process can't work properly.
It could also be a problem with the user data synchronization in the database. So check that the user account you're using on your frontend for your test is present in the Customers listing of HikaShop and contains the correct email/username.