i think its not a bad thing that she wants you to meet them, as long as she isnt bringing "bad" people into your life such as alcoholics, drug users, prior criminals, ect. there are many reasons she may be doing this- looking for approval, trying to "show" you how great a guy she met, push the relationship to a more serious level by involving families, or show you a person who is a better "choice" than her past ones. i dont think these reasons are malicious or anything. it just sounds a little immature. i think its reasonable to want to spend holidays and such with your current boyfriend. she may even want to show you guys off to him.
now all that aside, i think it is very bad she exposes her children to the different men in her life. you should never introduce your "serious" person every 3 months, her definition should be higher than "serious" when it comes to her children. she shouldnt involve her children until she is engaged!!! she is introducing her children to men she cannot really know in that short time, and may be bringing men with bad intentions into their life. she is constantly making her children bond with people that later disappear from their life. she is making her children think adult relationships do not last, and is teaching them a very negative outlook on life.
your only concern should be the effect this has on the children. unfortunately, from your description, i dont know if she is capable of understanding the negative impact her lifestyle has on them. if you and/or your husband mention it, i dont know if it will sever your relationship. if i were you, i would separate the negative behaviours from the annoying behaviours, and then address only what is truley a problem.
for the current issue, i would have her father discuss his hurt that he wont see his grandkids, and not focus on her life or his anger. when children are involved, you need to only address the big issues. good luck. it does sound like she is desperately trying to find someone. hopefully, she will find someone good, and she can finally start to be a grown up.