The alternator directly wouldn't cause the no start, but if it was not charging and allows the battery to not get a full charge it could. The electrical systems in the LS are very sensitive to battery health.
The rapidly flashing "Theft/Security" indicator suggests an issue with the PATS system. If you have another programmed key, try it. If you don't, double check all your fuses under the hood and in the spare tire well to make sure there isn't another one blown somewhere.
As for tracing down circuits from the wiring diagram, that's not something for the timid and does require some understanding of electrical systems and proper use of a DMM. If it's something you're wanting to try I can e-mail you a PDF of the wiring diagram for that circuit but it will probably be a day or two before I have a chance to save it off. Likewise with a diagram for the PATS system.
-Rod