In general a paternity action must be filed after the child is born so that the court can have jurisdiction over the baby. Typically while the child is unborned you need mom's cooperation to have the paternity test done.
As to the open adoption, that is something that must be agreed to by the adoptive parents as to how they want the relationship to be. Now it is possible for the biological parents
to permit their information to be released if the child ever requests it in the future, but for an open relationship will depend on whether the prospective adoptive parents are agreeable to this. So it would be something the parents would have to ask for.
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