As far as the nap goes, my 13 month old son sometimes takes both naps and sometimes not just depends on what he's doing when it's playtime with his 2 1/2 year old brother. Sometimes they play in their room until they both pass out and only take one nap at the same time but sometimes he will take both and the 2 1/2 year old won't take one at all and go down for bed a little early and still get up at the same time the next morning. It all depends on them. When they start fighting it then they don't need that nap.
As far as the pillow goes, I have one in both boys' beds. My 2 1/2 year old got one when he was about 16 months old but the 13 month old insisted on getting one himself at 11 months so what we did is we had one that got really flat because we'd actually had it quite some time and washed it so much that it's mostly flat now but not pancake flat and got ourselves a new one. Just with germs and the kids I usually wash it twice a week since we've had it for a while.