Remember me tick box appears randomly in my user/register page.

The remember me checkbox on registration is intentional under certain conditions, as noted in persistent_login_form_alter:

    // This is a user register form, but we only want to alter this if
    // - Visitors can create accounts and no administrator approval is required.
    // - E-mail verification is not required when a visitor creates an account.
    // - The form is not being executed by a user administrator.
    if (!variable_get('user_email_verification', 1) && variable_get('user_register', 1) == 1 && !user_access('administer users')) {
      $alter_form = TRUE;
    }

#765116: Integration with logintoboggan: don't show "remember me" on user registration unless "Immediate login" checked address an issue where additional logic is required when logintoboggan is installed.