Teething 2 Month Old?!

I know Google is not a doctor. But Googled "can a baby teethe at 2 months" and my son has most of the symptoms. I had my suspiscions when I noticed nothing soothes him except the bottle or paci with our finger in it. He has been fussy as Hell for 3 days, drooling (which he NEVER does), eating way more than usual, and had some red rash-looking patches on his face earlier for about an hour that came and went away out of no where. Has anyone else had a 2 month old teethe?