I use a baby cot for her room that she sleeps in at night. for convenience as well as her day time naps, she has a playpen that's in my room (cuz her room has no toilet) so her baths are in my room plus my parents' room and my room is on the same floor so easy for my parents to 'check in' on her as and when.
As for the living/dining room, we have a playmat which she plays on in the day (if she's in the living room) and naps on if there is anybody in the living room with her. I'm apprehensive about using a mattress cuz there's some height to the mattress and there's a chance she will roll and fall off the mattress
During her first 2 months i place baby in her stroller at home; mainly to get her used to being in her stroller and then i didn't have the playmat to put her on in the living room.
As for safety concerns, i believe that there should always be an adult in view of where-ever baby is at