This is great! The problem is, when you open the door and let her out, it's like going "outside" to her. You've let her out of her den and she HAS to go. So, now, just control her exit from your room. Either pick her up and carry her outside or put her on a leash and take her directly outside.I've been keeping her confined to my bedroom since i got her, and shes doing great in the bedroom. She recognises my bedroom as her home and never poops while shes in the bedroom.
Just realize that the bedroom is now her den and as soon as you let her out, she's going to go. Just manage that better. When she has eliminated and you bring her back IN, that's when she can spend a little time in the living room under strict observation.
As far as walking, 2 months is very young to be walking on a leash away from home. I didn't do that until my pups were maybe 16 weeks and had all their shots. rosmaryninja has given some great advice.