Try and remember that if you don't put him on the birth certificate, you've forgone your right to claim child support from him. The moment his name is registered with the Office, you can file for a child support assessment and he will be required to pay whatever his obligation is deemed to be.
Also, he doesn't have any rights. Neither do you. It's your child who has rights under s 60B of the Family Law Act, which says all children have a legal right to know, spend time and communicate with both parents on a regular basis, regardless of whether those parents are married, separated divorced or have never been together at all.
On the preschool matter, while ever there is no agreement or orders in place, dad is at liberty to do whatever he likes with the child, including picking her up from preschool one day and not returning her, so it's beneficial not just for him, but also for you to get some agreement in place.
I hope this helps in some way, and I do apologise if it isn't quite what you were hoping to hear.