So my ex-boyfriend left me when I was 20 weeks pregnant because I refused an abortion. His name is not on her birth certificate. Our daughter is now 2 1/2 years old. He wants to see her and take her out (at his convenience). We do not have a cusody order or visitation schedule in place yet. I have let him come in the house to be with her. He has seen her 4 times. He expects me to now let him take her out and away from my home. I am more comfortable with them establishing a relationship in our current situation as I do not know his level of commitment and trust with her. I give them time alone when he has been here. At this point in time, am I legally required to let him take her for an outing? Thank you!

No. Without a court order or custody agreement, you are not required to send your daughter anywhere. Let him sue for a paternity test if he wants to prove his relationship to her.

Vlee is absolutely correct, you do not have to let him near your child until the court orders you to do so. If he is serious about developing a relationship with you daughter he can petition the courts for such. But once he does so, plan for overnights in due time... it is almost impossible to avoid. Good Luck!