firstly it's worth thinking ( and expressing yourself ) in terms of what the childs rights are. The child has a right to two parents, has a right to a dad in thier life, has a right to be supported financially and emotionally by both parents. But that doesn't mean he can demand whatever he wants.
as a father your XP has 'parental rights' . This means he has a say in the childs education, religeous upbringing and medical issues. It also means that unless you have a residency order in place you will need to inform him and get his permission to travel abroad with your child.
To gain his 'parental rights' he does need to be on the birth cirt ( i think ) , and if it comes to applying to the CSA for child mainenance ( if he doesn't pay ) he'd need to be prooved to be the father. I have heard of cases on MN where the mum has had to pay for the DNA testings. I'd think carefully about not putting his name on the BC as it may be counter productive.
In theory, if he wants contact everyday, that is actually admirable, assuming he expects to do the nappy changing, feeding, putting to sleep, clearing up the sick, feeding, waking up at 2am, 3am, 5am etc etc.
That said, from your post, it does sound like he's using the contact as an excuse to be with you, rather than be with the child. Is he expecting to take the child to his house ?
I can give a little advice from the other ( male ) side. My XP told me I'd 'bullied' her into accepting my 'demands'. All I wanted was my dd every other w/e a visit once in the week and half of holidays ( fairly standard ) . What she saw as bulling I saw as sticking up for what I felt was fair. I can tell you one thing though, when my XP started saying that she would allow me to see dd, or that she'd let me have the time it infuriated me that she felt it was upto her to let me see my own daughter, so I'd advise you to choose your words carefully. You need somehow to get on an amicable footing with your XP, remember he may well be in your life for the next 18 years in some form or other.
Generally, and I'm only an armchair expert, if you disagree about access then solicitors will get involved, you'll be obliged to go to mediation, and then if it goes to court you'll probably resent each other for the rest of your lives. But it's unlikely that a newborn will be separated from mum for any length of time, but the courts will want to make sure the childs interests are foremost in any decission, and a child having a dad around is very much in the childs interests.