Oh Lizite,
You have to stop beating yourself up for a mistake you made on one night.
Firstly, your daughter is loved and wanted - she needn't know that you may not have had children - just tell her that you always planned children and that she came along quicker than expected. Talk to her about the good things of being a youngish Mum and the negative bits and also the good and bad bits about being an older parent. You sound a lovely Mum - she is the best thing that ever happened to you - that's what she needs to know and hear.
Secondly, her father made a choice when he had the one night stand with you. This was to use contraception or not. If he didn't then (despite what your idiot male friend says)he is equally as responsible for the creation of your beautiful daughter. It definitely does take two to tango but usually it's the woman who has to cope with the after effects.
Lastly, you are a decent human being who has coped with a massive life changing event. You deserve every bit of respect going from people and you need to know you are worth this respect. You have a daughter and you come as a package, believe me when I say there are plenty of nice, decent men out there who would happily accept this.
You have every right to have your needs respected. Your current bf might not be your daughter's Dad but you don't "owe" him anything. You are a good person who is worth treating well and more importantly your daughter needs to see that you are worth treating well as she will pick up her cues about how SHE should be treated as an adult from what she sees in childhood. Let her see what I can tell from your post - you are a worthy person, worthy of being loved, cared for, escorted home after a date and having your needs met in a supportive way. Dump this bloke if he really cannot give you that and find somebody who will - believe me there are plenty of decent men out there. I hope some of the Dads who post here will read your post - I suspect they may have an opinion on your current man.
Above all - stop beating yourself up. You are the Mum of a beautiful daughter who is the best thing ever to happen to you - you're both worthy of a decent man. You owe you man nothing for "sticking around" if he's not supporting you, loving you and showing your daughter that her Mum is a worthy person.