Hello,
Please use following css code to change the background of the mobile menu.
.mobile-nav {
color: #000;
}
And unfortunately, you can not show login link in the mobile in modern style.
Please add login page to the menu item in Appearance -> Menus menu.
Then you will see the login menu in the desktop an mobile.
Best regards,
Sorry,
Please use following css code.
.mobile-nav {
background-color: #000 !important;
}