I believe you have to set the name at the beginning when you first start the game.

After bath, click the clothes option (sock) then choose what you want, then click X. Now, grab the Sock icon and drag to the baby. You can also take clothes off by using the hand & dragging it to the piece of clothing you want removed. To check the diaper, when baby is on the table, take off pants with the hand icon by dragging it & dropping it, then grab the eye icon and drag it to the diaper & when the diaper lights up (becomes yellow), let go & you'll see the diaper