I'm not familiar with a rock and play, but would suggest you have a think about it's suitability for spending an extended amount of time in. Some baby equipment is ok for short use or shaped for a certain purpose but not suitable for extended time, for example a car seat is shaped to protect LO in an impact and many people leave their babies to sleep in a car seat once the nap has started but really it isn't good for their spine to be in long periods. Just something for you to think about.
With regards to napping in the living room. I planned to have DS in the living room for all day naps until 6 months (I was so worried about SIDS) but at 10wks DS had other ideas. He had napped great until then and suddenly wouldn't go down. After a day or two of problems I decided that, as he was still going down for BT well, I'd try him in his night bed for his nap, he stopped crying and went to sleep instantly! Amazing! Had I known then what Sara has told you I wouldn't have been so surprised but at the time I couldn't believe he was making choices like that.
If you have space for a travel cot in the living room it would double up as a great play pen later on when he starts scooting around the place and you just really need a safe place to pop him for a while.