First question: Just make the same form fields as the login form has. Here's the basics of it:
<form action="wp-login.php" method="post">
Username: <input type="text" name="log" id="user_login" class="input" />
Password:<input type="password" name="pwd" id="user_pass" class="input" />
<input type="hidden" name="testcookie" value="1" />
<input name="rememberme" type="checkbox" id="rememberme" value="forever" /> Remember Me
<input type="submit" name="wp-submit" id="wp-submit" value="Log In" />
</form>
If you like, you can add a field to the form to tell the login where to redirect the user afterwards, like back to the homepage:
<input type="hidden" name="redirect_to" value="<?php echo get_option('siteurl'); ?>" />
Bonus Question:
The $user_ID global will be set when the user is logged in and won't be set when they're not.
<?php global $user_ID; if ( $user_ID ) : ?>
Hello <a href="<?php echo get_option('siteurl'); ?>/wp-admin/profile.php"><?php echo $user_identity; ?></a> | <a href="<?php echo get_option('siteurl'); ?>/wp-login.php?action=logout" title="Log out">Log out</a>
<?php endif; ?>