In our last house, we had a screen door on the nursery. In this house, we closed the door to our bedroom--hated to do it, our old momcat (actually a male, now deceased) just loved us and all the baby's soft, warm things. He was also a "kneader", so in addition to the smothering concerns, we were worried about him scratching. He slept in the carrier all the time!
My sister got us a kitten right after kid #3 was born--he loved to jump into the portable crib. I didn't like it, but didn't get too upset because she was only in that when I was awake and downstairs, so I could keep on eye on things. It also had a cover, but we didn't bother. Now we just won't let any cats in our bed for #4--sigh! I miss the old momcat kitty who napped 23 hours a day--he was a great cat!