This turned out to be a hack. From what I have read, this hack adds back doors to all your domains and probably enters via a plugin. Yes, all of your domains. I thought I'd document some of my findings and solutions here in case anyone else has this problem.
Here were the clues that something was wrong:
-Feedburner feed stopped updating. Says feed invalid but feed validator says feed is fine. Nothing seemed to be wrong with the feed but Feedburner rejected it.
-Sudden, severe drop in traffic. My traffic decreased by over 75%.
-Rank on search engines decreased dramatically. In some cases from top three positions to page five or lower.
-Obviously, lowered revenue as a result of traffic and search engine rank decreases
Here are some of the signs of the hack:
-The first sign was that nothing worked to fix my feed and there didn't appear to be anything wrong with it.
-I found an unknown php file in my theme folder. The file was titled func.php. Details in my original post at the top of this thread.
-I deleted that file immediately and changed all my passwords for any account or user that can access my sites.
-Contacted my host to see if they could help. They sent me a list of files they found to have been hacked (I'll paste the list at the bottom of this post). These were all WordPress files in my index plus one plugin file. Note: I had already deleted several plugins, which may have also been hacked.
-These findings applied to all my websites under one FTP user. Which was pretty scary.
Here are the fixes (as best I can remember):
-First, I changed all passwords that access WordPress, FTP, databases, etc.
-Deactivate and delete ANY unnecessary plugins.
-Via FTP, I found unfamiliar php files on my domains with names like Siegfried_allegra.php, _cat_hyacinthia.php, and also some quickstart.html files. I deleted all of these. They appear to be the back doors that hackers use to get in.
-Next I removed the harmful code from all the php files for WordPress (see list of files below). This bad code was inserted at the beginning of those php files and looked a lot like the code snippet I pasted in the first post of this thread. Reloaded the fixed pages.
-Finally, I reset all my passwords again.
Then, I went to Feedburner and discovered that it now recognized my feed, which is a good sign, since that was the first alert that something was wrong.
Here is the list of files that contained hack code:
If anyone else has this problem, I hope this helps.