@Namechangeforthisone83 our posts have crossed about FB.
It sounds like she is clutching at straws by appealing out of the blue to his family, but if she is still at home at 31 then this approach will prolly make sense to her.
A paternity test is an absolute requirement. Your OH is not really in a position to wait until the baby arrives to determine how arrangements will be made, as he has immediate responsibilities to his children with you, you & the future of his marriage with you.
Given the OW’s behaviours & comments so far, the future looks messy & intractable. The harsh reality is that for your marriage to continue he has no contact with the child until a far far later time when the child can act & speak for themselves independently. It is a given that if the paternity is proven that he contributes to the support of the child.
This may seem harsh & unfair on the child not to have their father in direct contact in the earlier years, but it seems not too likely that the relationship between the parents will not be cordial or straightforward.
If there is contact between your OH & the child, then it needs to be somewhere neutral.
OP, this is your life, you get to decide & assert how your part is played. If you cannot get past various aspects of this situation, then that is how it is.
Your OH has to make some tough decisions about his life, & in full knowledge that he cannot have it all. He has responsibilities to other people & he cannot leave matters to see how it turns out as things go along.
No interaction until paternity test is complete.
If proven, financial paternal support formally negotiated & a clear cut regular support agreement made - no meeting to hand over the monies, no variation in payments.