Op, I've read through the full thread now, and with regards to your latest post- I think it's good that you've realised YWBU and selfish with the original funeral issue, and that you'll be encouraging your DH to attend.
I'm sure you'll be able to celebrate your birthday shortly after his return- it is hard to feel unappreciated/ unprioritised in a relationship (especially when couples therapy is exploring/ raising hard things to hear), but honestly in this instance there is no question that a funeral takes precedence.
However, for me, your posts about your SDD are hard to read. I know that your updates are only a snapshot of your issues, and perhaps you're not expressing yourself well, but the gist of it comes across as very derogatory and hateful towards your SDD. Your language and thinly disguised animosity towards the child is just plain wrong. For you to have laid down such strict initial rules about this child's involvement with your family life, really implies that you were not in a suitable position to enter into a relationship with her father in the first place. However, you are where you are right now, and you need to be making the best of it, even if that requires significant changes on your part. Your DHs 15 year old daughter is most definitely a child, and you just cannot blame her for issues in your relationship with her father. If cracks have started to show since she moved in, then I would suggest that those cracks have always been there/ not been caused by her. A solid relationship would be weathering the teenage storm together, not allowing a wedge to be driven between it.
FWIW, I have a 15 DSD, who has lived with us 50% since she was 1, and full-time since she was 10. She has a turbulent and emotionally damaging relationship with her mother, who she has fortnightly 2 hour contact with.
Accusations a-plenty on both sides, tantrums, outbursts, total breakdowns have ensued since puberty, but I still love the bones of that child. Her destructive and hurtful outbursts towards her father and I could so easily be written off as deliberate/ manipulative/ bratty behaviour, but the reality is that she is a child who has been emotionally damaged from a young age, and who at times of change or uncertainty will push against us in order to test if we will also let her down/ get rid of her, as has been her experience of the world so far. It is hard-going, intense at times, frustrating and tiring, but that is the job of a parent/ step-parent that I signed up to, no matter what were the original arrangements when I first got with my DP, or what I envisioned our future would look like. Her unsettled childhood and damaging family dynamic is our job to help her navigate through. She is my family in all but blood, and I will nurture, love, protect, support and simply hold her when all else fails, until such a time as she is old enough to deal with the complex issues and grievances she is battling with, in the hope that she comes out the other side as unscathed as possible.
As well as attending couples counselling with your DH, if you are serious about being with him and his daughter, then you should both be arranging counselling for your DSD to help her with her obvious unhappiness, building your relationship with her through time spent and patient effort given from you, encouraging your DH's relationship with her one-to-one and with the whole family, making sure that you present a united discipline front with your DH, involving your wider family with her to show she is a loved and much wanted member of your nuclear set-up, fostering a wonderful relationship between your DS and DSD to make her realise the simplistic love and adoration of a younger sibling, engaging with her on her hobbies and actively sourcing further similar interests to broaden her horizons, helping her to apply for further education and weekend job settings in your local area so she knows without a shadow of a doubt that her future is assumed and very much wanted at home settled with you and DH.
I wish you luck for the future.