Add the Date to last Updated field on Available Updates Page

You should use format_date() instead of date()

+  $output = '<div class="update checked">'. t('Last Checked: ') . format_date($last,'medium') . ($last ? t(' (<b>@time ago</b>)', array('@time' => format_interval(time() - $last)) ) : t('Last checked: never'));
+

1. In D7 the string concatenation is done with a space before and after the .. Even before D7 you need to have a space after a , (take a look at the format_date parameters).
2. The string inside t() it should not start or end with a space.
3. You should follow the example you already have in the code you change and append the date as the time ago is. This way you will also get rid of the error in your patch that makes a "Last Checked: Last checked: never" message appear when there is no value for $last.
4. One too many new lines.
5. Why do you change REQUEST_TIME to $last?

Please post your new patch here: #356841: Update status 'last checked time' should output actual date of last check.