Is there a way to make an alert action send a crafted syslog packet? How?
The simplest way would be to create a script, and then use that script as an alert action (see http://docs.splunk.com/Documentation/Splunk/6.3.3/Alert/Configuringscriptedalerts ). I'm not sure if you want to simply write to syslog or try and send to a remote syslog server. Have a google, there are many ways to achieve those things.
If you want to make a reusable action have a look here: http://docs.splunk.com/Documentation/Splunk/6.3.3/AdvancedDev/ModAlertsIntro