Welcome! I'll preface by saying I can come across as crass and blunt, so take what I say the nicest way possible:

• The XW/BM doesn't have to like you. She doesn't have to get to know you. You're dealing with a high-conflict ex, and honestly, the less you have to interact with her, the better. With someone like that, you don't want to try and form a bond. She'll only use it against you.
• Your DH needs to put in some boundaries. It's hard; I just blogged yesterday about how my DH still struggles with it and we wonder where a good cutoff is. In the near future, he can just ignore her texts that don't pertain to the child. If they become too frequent or harassing, move to blocking her from his phone and switching to email-only communication. Yes, she'll fight at first, but hopefully she's a decent mother and won't let it interfere with your DH's parenting time.
• Do separate birthday parties. It won't traumatize the girl, and she'll still get time with dad. Just do separate all holidays. This gives your SO the chance to create new traditions and memories with his daughter.
• Go to SD's public events. Don't sit with BM. If she throws a fit in public, it makes her look bad, not you. Don't sit back just because she whines about it. That gives her way too much power.
• Your SO needs to address your needs above his XW. If her texting bugs you, he needs to put a stop to it. No woman will put up with that behavior long-term. If he knows she is toxic, then he needs to cut her off. And THAT needs to be a requirement before you get married.
• Your SD probably has conflicting feelings about you that stem internally and externally (i.e. BM). So long as you are comfortable and she is respectful, be kind and have fun. When she needs space, give it. Her behavior sounds normal. Remember, no one can be happy all the time.
• Make sure your SO actually parents. Weekends can be fun, but there should be structure. If SD is misbehaving, she should be punished by dad. She should have a bedtime. She should have to shower. She should eat home cooked meals and not just junk food. Fun, Disney parenting works until it doesn't, and when it stops working, it's catastrophic failure with lazy, entitled kids who won't listen to authority or show respect. Dad is a parent first and foremost with his kid - good, bad, and ugly.
• Remember, the XW is NOT your ex. It is YOUR SO'S JOB to reign her in. He is the one that had a child with her, not you. She is NOT your mutual responsibility; she is solely his. Do not get dragged into the middle, but also don't let her needs and wants trump yours. If your SO wants to treat her with kid gloves and make her happy, he can get back together with her. You deserve BETTER and MORE than her. Do. Not. Settle. For. Less.
• Never, ever feel like you can't tell your SO how you feel. Communication is a necessity. You cannot be afraid to talk about these things with him, and you can't be afraid to expect better. You need to be treated like THE wife, not the SECOND wife.

Do not make the mistake of marrying this man come Hades or high water. If you are uncomfortable, that will only change if he makes changes with his XW. This isn't something you can or should fix. Loving him, and him loving you, isn't enough.

When I started dating my SO, he and his ex had a very close relationship.  She wasn't needy or crazy, but they were very friendly and there were no boundaries.  I told him there would be boundaries or I'd be gone (and I meant it). He complied pretty quickly, as he did not like the consequence for not doing so. They still do joint birthdays, and they're still friendly, but it's nothing like it used to be.  

If you do not like the relationship he has with his ex right now, you need to speak up and demand change. I would not marry him until you've seen change. It will not magically get better once you're married (it'll probably get worse).  If you don't want to be the 3rd wheel in your own marriage, address this NOW. 

Your SO needs to put boundaries in place... His ex shouldn't be able to be dependent on him, or frankly (especially in high contact situations) be able to contact him for any purpose other than for it being specifically about the skids. What she's doing is basiclaly like still being married emotionally to him, and that's NOT healthy for your relationship...

From experience... An ex that call son non-child related things can hurt you... Especially when it's sob stories and your DH is good hearted and "feels bad." It can build resentment and hurt and cause you to question YOUR place. She needs to understand it's not her place, and your SO needs to enforce that or you're in for a world of pain.

Seperate birthday parties are a totally acceptable thing, she'll still get special time with her dad, it just won't involve the crazy woman and will probably be better for the skid anyways, especially when BM seems to think she's still married to your SO. I've had moments where I've literally wanted to (nearly did, there was also the time I was holding a bat... It was a close call) deck BM for flirting with my DH (at least trying), refering to me as "second wife" and my DH as "her former husband" (not even using the word ex), scooting close to my DH, shoving me, trying to get DH to be HER emotional confidant, showing up uninvited, cussing me out (she's lucky... I do kickboxing... I was seeing red), etc. Basically acting like she still runs the roost. 

It drove me mad, I was crying nearly daily from the stress, felt like my stomach was upset a lot, and I was living on edge wondering when she may end up jumping in again, and when I was going to be made to feel like a second place spouse again... So I drew a line. DH and I had a chat, I explained the situation, and things are going MUCH better now. (also helps BM kind of poofed). But your SO needs to create boundaries now and expectations, or he'll never get them in place and you'll always be secondary... That's not fair to you. Your SO needs to solidlfy YOUR place as his SO, and make sure the only place she has is when it comes to the skids and info and decisions regarding them.

he feels it's best for the child if he just let's it go and keeps the peace.

That's a very common excuse and I'm sure some men believe it when they say it, but it's an excuse just the same. They don't want to go through the storm, and I get that, but your fiance is trying to make TWO women happy and it's just not going to work.

He thinks it's what's best for his daughter but he is allowing her to be used as a pawn and a manipulative tool by his ex.

You don't need BM's approval and it's best to keep her at arms length, since she is high conflict, but your fiance should be keeping her at arms length, as well.

I would not marry him until he can prove that he can put, and keep, some boundaries in place. He is still emotionally enmeshed with his ex.

Your STBDH needs to stop responding to her texts about things that don't pertain to SD.  He also needs to cut her off when she starts talking about things that do not pertain to SD on the phone.  She is still emotionally involved with him and he is exacerbating the problem by allowing it to continue.  If you keep feeding the monster, it's going to keep coming back for more.. YKWIM? 

It is hopeful that you STBDH took a stand and refused to go to party if you were not invited.  That is definitely a step in the right direction.  While BM is being a petty beeotch here, as LDad said above, she doesn't have to like you.  Just be the better person and don't let her know that it bothers you because that gives her power.  Just go to whatever games, concerts etc that you can and support SD and ignore BM if that's the way she wants it. 

Hopefully she'll get a boyfriend soon and move on from your STBDH.  Once they get a new man in their lives, they usually tend to tone it down a teensy notch with their dependency on the ex. 

Lots of good feedback here so far, but the takeaway is that you should not be marrying until your SO ties up all the loose ends of his last relationship. Are you doing premarital counseling? 

You are getting good advice. There can be only ONE wife. And that is not his ex. If he does not begin to put boundaries in place, you will indeed be second whenever she has an "issue." Let's say you and DH are working on a project--painting the bathroom maybe. All is going well until his ex calls and she needs him for something--ran out of gas, dryer isn't working, etc. In order to "keep the peace" and "for his daughter" you WILL be abandoned so that he can take care of whatever issue the ex is having.

In his man brain, he avoided a conflict with his ex (yay!), the issue is fixed so the daughter has no negative impact (yay!), and the project continues because you are competent and capable of picking up his slack (yay!). You, on the other hand, are resentful as hell because you were counting on his help and the project was an important thing for the two of you to do together. You will feel less important than the ex, and he very conveniently won't understand why you have an issue with him helping her out "for the daughter."

Well, it's not for the daughter. It's for HIM. He avoids a conflict and gets to be the hero. The hero to his ex. A jerk to his wife.

Boundaries. Now. This won't get better -- it will get worse.