Dear T.,
The rash is probably unrelated to the formula, an allergic reaction would be a rash around the mouth or a diaper rash. I wouldn't worry about it. Its probably an environmental thing, just wash the area with some mild soap and water.
I'm wondering why you went straight to soy formula/lactose free? Animal protein is the easiest to digest, so you really should try to supplement with cow's milk based formula. If you want to build up your milk, try pumping between feedings.
Everything considered, it sounds like you should consult with a lactation consultant and have them check to see if baby is latching correct. You are probably producing enough milk, remember babies this young, eat often and if you don't have a boppy pillow already, go and get one. It really makes a difference.
Congrats on your new baby!
N.