Talk about stabbing you twice. Anyway, this does not show a lot of inclination on his part to be suportive of you. Your OW must be a bit nutty. Putting your H name on birth cert. is strange. YOu may want to try and find out law in your area on that.
Web is good for research.
Anyway, she is not thinking about her other child to do that. They should both be raised together. But I know you have no control over that. You don't have to make anything public, now or ever. If you do decide to make it public, make sure that you have DNA testing done on child first to determine if it is his.