OK, I found the solution for the custom CSS. The reason why it did not work before was because I was using 'get_bloginfo' instead of 'get_stylesheet_directory_uri' (I have a child theme). Meanwhile, my customer changed things around and now he wants a custom login page all the way which means that not only a new CSS sheet will be involved, but also a new markup (I will need to create more nested divs and classes). I understand it is not reccomended touching core files but this cannot be done by simply editing function.php. I heard it is possible to create a static page and then insert the login form into it but I don't know the correct procedure.
Please let me know if I have to open a different thread since here I was requesting a solution only for the first part of the issue.