Hi,
Joomla 4.3 changed the way the form validator system works on their end. It's a javascript file which is responsible for hightlight the border of input fields in green or red when the user is filling them.
Before, it was possible to target a DIV and now it's only possible to target a form element.
Because of this, we had to change the line:
var container = document.getElementById('hikashop_checkout_login_<?php echo $this->step; ?>_<?php echo $this->module_position; ?>');
to:
var container = document.getElementById('hikashop_checkout_form');
in the file components/com_hikashop/views/checkout/tmpl/sub_block_login_registration.php
When I press the login or next button on your checkout while the login form is displayed, I can see these error messages in the console of my browser :
i.imgur.com/VeZpWEl.png
These error messages are specifically for the input fields of the registration form, so I'm feeling these are the reason why the form is not submitted on your website.
Now, when I try to reproduce the problem on my local test server, I do not get the error messages in the console and the login form is submitted, even though I still have the same change as you. So the problem is not just with this code change, and it might actually be unrelated.
And since it works on my end without changing anything, the first reason which comes to mind would be some kind of issue with the javascript of the template you're using, but looking at your pages code, it seems you're using Cassiopeia, the default template of Joomla, like me on my test server.
So basically, I'm not sure what's going on.
You could first try reverting the change I talked about above, to see if it helps.
Regarding the buttons question:
The login / register buttons are displayed by the "login" block you have in the step of your checkout workflow. There is actually a setting there to hide the "register" button.
The "next" / "finish" button at the bottom is the main button of the checkout system which allows the user to go from one step to the next one.
While both buttons can potentially do the same thing, it is not necessarily always the case. For example, if you have a step with only the "coupon" or "cart" block, you need that "next" / "finish" button to be able to go to the next step.