Could the heater core cause this? Hose going to the core? First thought of a hole in the floor, but it has been dry for a week here.

 

If it was from the heater core or a hose you would smell antifreeze.

There are several things that could cause this. Do a search and you will get all sorts of information.

 

It's likely your a.c. condensate drain, it's a known issue with a $2 fix. Search that and you will get a detailed repair thread

 

My lis wet when it rains. Figured out it was the plastic door step. It has clips on the bottom and the clips go through the bottom of the truck which rain can go through

 

If yours is wet only when it rains, you probably have a leaking windshield. I seriously doubt the water is migrating UP through those clip holes. Everybody's truck would have the same problem if that were the case. You're finding water down there in the trough because that's where gravity is taking it.
To the original poster, look at your evaporator drain tube where it exits the firewall. Its a short little stub and years ago Ford came out with a service bulletin wherein they installed a 90 deg. elbow/extension to cure exactly the problem you're having. My 05 never had a problem until I installed a 2" leveling kit, and then my wet carpet began to appear.

 

Third brake light is also a possibility, water makes it way down to the door channels. Just had to reseal mine with some rubber silicone.

 

Wow i installed a 2'' leveling kit also, so your probably on point here then.