Getting URL Variables with a Rewritten Page (Login Page)

If you want instead of redirect to wp-login the user redirect to login page then use the following code. It will redirect user to login page where login form is present.

add_filter('site_url',  'wplogin_filter', 10, 3);
function wplogin_filter( $url, $path, $orig_scheme ){
    $old = array( "/(wp-login\.php)/");
    $new = array( "login/");
    return preg_replace( $old, $new, $url, 1);
}

add_action('init','possibly_redirect');
function possibly_redirect(){
    global $pagenow;
    if( 'wp-login.php' == $pagenow ){
        wp_redirect(site_url( $path="login"));
    exit();
    }
}

In login form ckeck the value of action. For login use wp_signon() function.

if(!isset($_REQUEST['action'])) { ?>
      <div>Login with the Email
        <form id="login-form" action="<?php echo $_SERVER['REQUEST_URI'] ?>" method="post">
          <table style="float: left;">
            <tr>
              <td><label>User Name</label></td>
              <td>
                <input type="text" name="user_name" id="user_name" placeholder="Enter your username">
              </td>
            </tr>
            <tr>
              <td><label>Password</label></td>
              <td>
                <input type="password" name="password" id="password" placeholder="Enter password"><br>
                <a href="https://wordpress.stackexchange.com/questions/104241/<?php echo site_url("login/?action=lostpassword') ?> "> Forgot Your Password ? </a><br>
              </td>
            </tr>
            <tr>
              <td></td>
              <td><input type="submit" name="submit" id="login-submit" value="Login"></td>
            </tr>
          </table>
        </form>
<?php
}

For lost password. Create form in the same login page

if(isset($_REQUEST['action']) && 'lostpassword'== $_REQUEST['action'])
{
  ?>

  <div id="tab3_login" class="tab_content_login" >
   <p>Enter your username or email to reset your password.</p>
   <form method="post" action="https://wordpress.stackexchange.com/questions/104241/<?php echo site_url("login/?action=tg_pwd_reset') ?>" class="wp-user-form">
    <div class="username">
      <label>Username or Email : </label>
      <input type="text" name="user_login" value="" size="20" id="user_login" />
      <?php toolTip('user_login', 'Username');?>
    </div>
    <input type="submit" name="user-submit" value="<?php _e('Reset my password'); ?>" class="user-submit" tabindex="1002" />
  </form>
</div>
<?php
}