Automatically add “new” icon on new posts

Sure. In the Loop, you need something that will get the time the post was written, and then compare it to “now”, then act on that difference.

First, we get the post time.
$post_time = get_the_time('U');

Next, we get the time of last week (7 days):
$last_week = time() - (7 * 24 * 60 * 60);

Then we compare the two and output an image if the post time is after that last week time (meaning that it’s newer than last week):

if ($post_time > $last_week) {
echo '<img src="youricon.gif" />';
}

Simple, huh?