I'm so sorry that this has happened to you. He sounds awful, and the new girlfriend had no business messaging you like that!
FWIW, DC1's Dad had walked before he was born, and turned into an absolute tool. He also moved overseas. 3yrs later he contacted me and asked about being involved. I saw a solicitor, who said that he could be added to the Birth Cert, and told me the procedure for this, but advised me to be cautious about allowing this in the first instance. I told XP that I'd allow contact only if he was polite/civil to me, and if he actually meant it and would stay in regular contact (as I didn't want DC1 to be hurt and messed about). I told him that I wouldn't consider having him added to Birth Cert until he'd proved beyond doubt that he was in this for the long haul. He not only started paying child support (every month, without fail, including enough to cover the 3yrs previously), he also apologised (to me, and our family and friends) for his previous behaviour towards me and DC1, and has remained in frequent and regular contact ever since. After 3yrs of this, we had further talks and only then did I have him added to the Birth Certificate (he had to come with me for this). We co-parent well, he's polite and respectful. DC1 and him have a very close relationship, which is great.
In your situation, please please register the birth alone (i.e. don't let him come with you and have his name on the birth certificate) - I'm assuming that you're not married (if you were married, he'd automatically go on Birth Cert and have Parental Responsibility, even if you registered the birth alone). This isn't just a thing to just have "Dad's name on birth cert, and oh how nice that would be". You would be handing over equal Parental Responsibility to him (and he's shown that he's unreliable, unpredictable and behaving despicably). He could veto the school you send your child to, things to do with DC's health, it would mean he has a legal right to pick your DC up from school or childcare, he could veto travel/holidays abroad, and so on....all sorts of things, all through your DC's childhood and into teenage years.
As for contact, overnights wouldn't be in baby's best interests whilst newborn/so young. Contact would need to be built up to that.
I know things are scary, but you will be ok. Take it one day at a time, get support from family and friends, and don't let him upset and intimidate you into doing things you're not comfortable with (i.e. being at the birth etc). You do what is best for you and baby, and try to keep contact with him to text/email so that you have a written and time/date stamped record of it. Keep screen shots in case he tries to delete stuff. Wishing you all the best x