When the oldest SD started getting out of control, I told SO that you teach people how to treat you.  You don't have to put up with anyone being rude to you, especially not a child.  He needs to get in that a$$ and if he doesn't, you should.  I also want to be a good partner but my SO's oldest is not going to use and abuse me.  Luckily for me, SO has my back all the time

That attempt to pick up this kid would be my last.

Let Dad and BM take her and pick her up. With the snot attitude the kid gave you, not even a sincere apology would make me change my mind. Nope. 

I think it's good she is going. Great BM paid for it and didn't ask Dad to pay half ( maybe they can take turns paying every other year?) and that the kid didn't just sit around expecting to be feed and entertained at home. However, that attitude of hers would be my boundary line. 

No it is not pregnancy hormones. The kid was a butt head. Luckily for you, she's not your butt head and you do not have to tolerate such treatment from her. 

And no, I would not allow her to have any guest (not even your cousin) over until x ( fill in your own time penalty) And yes, I know the other girl is only there until like July 4th. Doesn't matter. She was disrespectful directly to you and this one is yours to hand out the consequence. You. Not Dad. She didn't shake attitude at Dad, she shook it at you while all you were trying to do is a favor helping Dad and BM out. 

"SD if you want me to be nice to you and do things for you stop being a brat. And you can tell your dad and mum whatever you want. But you are rude and sullen and ungrateful. If you were polite and thankful I would have driven over to get your cousin back. But you copped a shitty attitude so this is what happens. And the longer this continues the harder it will be on you. Your Dad wants us to get along. However there is only so much I will do before I give up. I will not be picking you up tomorrow because I dont want to. I do not care what your father says, I am not your personal unpaid Uber driver for you treast like crap. So instead of coming home to a nice cool house you will be stuck at day camp until one of your parents collects you. Your call. Either be nicer or face the consequences."

dad

I'd have probably left her a$$ there until she called me and apologized.

"Dh tells me to build a relationship"   WTH!!!! No wonder she is a snotty brat. Sorry, but you dont build relationships with rude disrespectful skids who appreciate little. I really dont know how you do this, or any of the ladies on here that have to deal with skids daily. No words of advice, just empathy. Adult skids are hard enough, but they arent daily interactions thankfully.

I would take the advice of the above posters, no more chauffering her until her attitude changes. 

Therapy is always a good idea. Go by yourself - do this for you.

In regards to SD, I think her age is partly to blame, then there’s the parenting, or lack there of, from BM and your DH. Parenting is a both a skill and a balancing act. It’s a lot of selflessness (on the parent’s part) and heartache. 

I think your intention is in the right place, and BM and DH should be so lucky to have a SM in their child’s life like you. I don’t know. I think I would have been long gone, with the lack of respect and support from them, given all that you do.

I think you have some self-reflecting to do, in regards to what you’re going to engage in and not. Perhaps journal these thoughts so you can go back to them as needed. 

YOU need to be consistent for your own sanity. Your DH needs to be a better communicator, and BM needs to stay in her own lane when it’s not her time with SD. 

I tell him that she hung up on me and was mad that I didn't want to drive across town and back to get my cousin so she was being rude. He tells me to "build a relationship".

Really? Why does your husband let his daughter get away with being rude to you and hanging up on you? My DH would have had my SD's ass.

You want to be a good partner and help your husband out with HIS daughter and his reaction to her acting like that, is to tell you to build a relationship? I know what I would tell him.

If that would have been me:

'Listen, you little b!tch... You want to act like a b!"tchy little sh!thead, you go for it - but I`m done. I don't want to be around such a nasty girl. I'm done giving you rides, I'm done cooking for you, I`m done buying you things, I'm done doing nice things with you - I'm done with your attitude. You're not my kid, I don't owe you anything... and I'm not going to try and do nice things for you just for you to be a smart-assed little b!tch. When you decide to knock off the attitude and stop being such a b!tch, maybe I'll start doing nice things for you... but no more. Just by the way, I was going to treat you to McDonald`s and ice cream, but since you`re being such a b!tch, I'll just go alone or with your dad, and you can stay home with a crappy sandwich... and as for tomorrow, you better tell your mom to come get you, because there`s no way I'm putting myself through the torture of being around a little b!tch like you when your dad`s not home. Now get out and WALK.'

I've done something similar with my middle daughter when she ramped up with a little smarting off last year.....I made it quite clear to her that she was NOT at my level - that I`m up here and she`s down there... I`m the boss, she`s my b!tch... and that I can always choose to ship her off somewhere else and just pay the basics for her needs....that if she wants to stick around, she better start playing by the rules...haven`t had an issue since.

Look at it from this view:

You went from EOW to 50%.  So did SD.   She is used to having her Mom 26 days a month.  She is used to her Mom being the one picking her up and asking about her day or just having compatible silence.  She is used to being a little bitchy to her Mom and its OK bc Mom loves her unconditionally.   She is used to Mom and knows what Mom expects and lets slide.

Now half the month she is with you and Dad.  I'm sure she senses your tension, she is going thru a lot herself at this age, so she has a lot goig on in her head right now.

Sit her down and talk it out, or at least explain your feelings and anxieties and try and build a working type of relationship with her.  It doesn't have to be rainbows and unicorns, but a coexistance every other week.  When you have your baby, there will be times you may need her to help out a little bit and she may really want to help you with the baby.

I'm in no way taking SDs 'side'.   I went thru this with an angry 9yo taken from her Mom, didn't want full time, didn't want to raise kids again but I had to either come to an understanding with her or break up with DSO. While we have no love lost, we were able to have a peaceful coexistance, I had to be the one running her places and helping out at times so I had to be a big girl and at least attempt to work it out.

Today, I don't give a crap if I every see her again, but she is with BM and stb18 so I can be a bitch

Yeah, that “build a relationship” comment is too much. 

My DH would literally never say that to me, knowing I would kick his balls up into his throat.