These are things you need to discuss with her and maybe even the extended family now, don't wait hoping it will get better. These are things you've already discussed so they should have been clear boundaries to follow. I think you need to be firm in the boundaries you've already set and don't back down (unless there's something your willing to compromise on).
Right now your setting the foundation for the future of your relationship.
All of the situations you've listed (her signing her name as mom, birthfamily using a different name than agreed upon, birthmom inviting birthfather to visits, etc). Would all be major concerns for me as well.
IMO the bottom line is they are not respecting you and your DH's decisions as the parents of this little girl and if you don't say anything and set some clear boundaries, I think your looking foreward to a very diffiuclt relationship with them in the future.
(I understand they are going through a rough time, 3 months is really a short amount of time to process everything. But it will be much easier to correct this now and build towards a better future than let things slide until it all goes down hill and then try and dig yourself out of that ditch later on.)