Hi Nooshie, I don't know all the facts but work at a law firm so PM me if you want the number of good lawyers
Yep it's called a Parenting Order now. Because he's named on the birth certificate, he does have rights but that doesn't mean he can take him wherever he wants or do whatever he wants with your son.
You can get an order in place for supervised visits, or visits at say his mother's house where your ex has to stay with the child. Also you can stipulate who they are NOT allowed to put the child in contact with etc.
I've recently looked into this due to my own circumstances and it seems pretty straight forward if you get a good lawyer who knows what they're talking about.
I have heard, and don't know if it's true or not for couples that haven't been together while the child's alive, but sometimes (for couples separating etc) they can order counselling to determine childcare arrangements etc. (free through the courts).
you may find, however, if the father's contact is rather sporadic, you stick a motion to get a Parenting Order sorted on him and he'll back off (whether that's a good or bad thing only you can decide).
Oh and most of the time it's done through the lawyers and just signed off by the courts - very few I see are actually taken to Family Court to be disputed etc as they mediate first.
If you're worried about him drinking around the child, or putting your son in situations where you wouldn't feel comfortable him being in control of your son, I would document such situations in case you need evidence to back up your application.
Good luck