How is the new husband being "very disrespectful of the situation with my fiance"?

It depends on the mom's attitude toward you. I would stay out of it if she is not nice (for now) and tell your fiance that if the new husband is being disrespectful, to just be cordial and not to get pulled into an exchange of words or worse. It isn't good for children to be in a situation that is hostile and they feel like it's all because of them. They blame themselves for the fighting. You might try to befriend the mom and let her know you'd like to be on the same side. I'm assuming that everyone has the same goal to raise a happy, healthy child. If you get married, you are going to be in this child's life, at least for purposes of raising her, for the next 16 years. It doesn't end when they turn 18 and if there isn't some kind of joint effort (on mom & dad's part) to come together for what's best for their child, then they are going regret it.

As for implementing a relationship with her?? I would suggest again that you put aside your opinion of her mom and at least TRY to befriend her. Mom's can sometimes feel threatened that someone else is going to try to take over their role. He dated her for 4 years so he must have found some good qualities in her. He had supervised visits so she must have been doing an ok job and is sure to be concerned because this child is 2. I would be nice to the mom, talk to her if she is willing and let her tell you how she wants her child raised. Then do your best to abide by her wishes. Unless they affect the health & safety of the child, then the best way is to have the mom's support of your relationship with her child. If she feels threatened by you (in her child's life) or becomes jealous or whatever negative feelings, the child will have feelings of guilt if she likes you or she may choose to resist liking you, taking her mom's side.

I have said it over and over, and i hope this helps, TAKE A COOPERATIVE PARENTING CLASS. We did and even if mom is not receptive or cooperative, it gives you and dad tools on how to avoid being pulled into battle. Hopefully she can make her decisions based on what's best for her child, which is to have a loving relationship with both parents.

Why were his visits supervised? That is not the norm.

First of all thanks for your responses. I really appreciate the feed back. As to the way that BM's husband is disrespectful: is that any time my fiance goes to drop his daughter off the BM's husband will grab the daughter from him before he can say good bye and then try and stare him down, basically trying to play some kind of dominance game with him. My fiance never responds to this kind of behavior, but I know it bothers him. As to the supervised visits...the BM tried to disappear when she was still pregnant (she basically was using my fiance as a sperm doner at that point) but my fiance really wanted to be part of his child's life. By the time he finally tracked her down he established paternity through a lawyer and then his lawyer left on vacation. In the mean time he hadn't seen his child at all and she was getting to be about 4 months old. At that point he couldn't wait for his lawyer to come back so he signed whatever her lawyer came up with simply so he could see his child (he was working off emotions at that point). Basically her lawyer made it so that she had control of everything and would be present at the visitations. So that pretty much sums up why he eventually went and fought for joint custody. It was his right at that point to be able to see his daughter without having to see the mother.