Re: "Baby acne"
I think you don't go anywhere with this, for the time being. 5 week-old babies have very delicate skin, they tend to get normal baby acne and various diaper rashes... Everything you're seeing could be totally normal, and not a symptom of anything. Rather than going down the rabbit hole of dietary eliminations, just keep observing your baby for a while longer. If he seems happy and healthy and is growing normally and not having weird poops, then this is probably either nothing or something very minor.
Coolest thing my big girl said recently: "How can you tell the world is moving when you are standing on it?"
Coolest thing my little girl sang recently: "I love dat one-two pupples!"