As MollieO said, if the father isn't on the birth certificate then he won't get automatic Parental Responsibility. However, he can subsequently get PR by either agreement with the mum or by applying to court for a Parental Responsibility Order. Courts rarely refuse such requests as they regard it as the right of a child for both parents to be involved in their upbringing unless there are very good reasons not to.
Assuming there is no Residency Order in place (ie, they haven't gone to court to decide who the child lives with) then if/when the dad gets Parental Responsibility then neither of the parents are allowed to remove the child from the country, even for a holiday, without the other's permission. (Yes, I know lots of people do it and get away with it, but strictly speaking you're not supposed to and the US immigration officers can be sticklers for the rules).
He could apply to court to prevent the mum taking the child on holiday out of the country but it's unlikely a court would agree unless he can show reason to believe that they won't be returning (eg, she's got a one-way ticket, or has left her UK job and is selling up). If so, he could apply to court for a Prohibited Steps Order to stop her. If there's any suggestion she's not intending to come back then the court will likely stop her (at least as a temporary thing) until CAFCASS can investigate, produce a report and the court can hear the case properly and make a considered decision. That can take months.
When two parents both have Parental Responsibility, they are also expected to discuss important matters such as medical care and schooling.
Maintenance is dealt with entirely separately and is unaffected by Parental Responsibility or names on birth certificates. If a mother names a man to the CSA as the father of a child, the CSA will expect that man to either pay the calculated rate of maintenance based on hius income or to prove that he's not the father through DNA test. I'm not sure how much the CSA pays attention to assets rather than income, though - all the basic information about the CSA I've seen shows maintenance calculated as a percentage of income.