Is there a way to login by directly sending a command to index.php?
I suppose, it could be possible, but it should be a POST request formed manually. Just check parameters in the source of the login form, you should send user_usr, user_pwd and mode values, then index.php will check them in database and will set (or will not) the proper cookie. By default, it is not possible to send a GET request like you propose, this is a question of the customized script, which I suppose, is also quite possible and not difficult to program. So for example, you may code the script named login.php and organize its input and output in the way you need.
As about reading login from the memory card, I must say "sorry" because I'm not familiar with such technologies, never seen that stuff and could not give any precise advice.