Changing Log-In function

It depends the registration mode you're using. If you're using "no registration", no joomla user account is created for your customers when they register, so they can't login.
But if you set it to "simplified registration", they will get an email with their username (their email address) and their password (automatically generated) and they will be able to use that to login onto your website.

The style of messages are handled by your template CSS.

Then you should use the "no registration" value of the option "registration" of the checkout tab of the configuration.

It's not possible to just have the users logging in with just their email as joomla doesn't allow that. You have only three possibilities:

1. Normal registration: users have to enter a username, email and password when registering. They can use their username and password to login afterwards.
2. Simplified registration: users only have to enter their email when registering. Their email will be used as username and a password will be generated for them and sent to them via email. They can use their username and password to login afterwards.
3. No registration: users only have to enter their email when registering. No user account is created for them in joomla so they can't login afterwards.

The login page is indeed supplied by HikaShop. But behind the scene, what HikaShop does is that it calls the joomla login function with the username and password your provide. HikaShop is just in between but does not control how joomla behaves. It's not that we don't want to help you, no, it's just that it's the way joomla is done. I'm here to help you but I can't do the impossible. It is impossible to login someone to his joomla account with just the email address. The password is absolutely necessary. It's the way the joomla user login system is done and there is no work around besides rewriting the whole joomla user handling.

I'm interested to see what you find because I don't see any easy way to do what you want :/