Sorry for not getting back sooner, didn't have a moment to read and reply. Just got back off holiday. So to answer the questions:
Me and DP do not live together and have been together 4 years. I've know DSD 3 years.
Me and DP don't share any children. DS is my child and is younger the DSD by a few years.
The ice cream situation as stated it was a multipack for the holiday home. I'm absolutely fine with choosing a couple different ones for people as we might not all enjoy one. It's the fact she flung open the door, grabbed them and shouted "I don't care about other people, I'm having these" without a discussion. DP must have promised her them back home which I was not aware of until she then said it in the shop but DP completely dismissed the rude behaviour at the time. I just picked this one example at the time. There are many many different ones.
The crisps situation, I didn't offer because I'd barely had a look in because DSD had already eaten the majority of them so I decided to have some. It's like she was territorial of them even though the food was for everyone.
Other examples of the past few days were DP and DSD wrote in the guest book. DS wanted to write a little thank you too so he did so this morning. DSD saw he had done this and then proceeded to wrote something else in the book straight after him and much longer even though she had already done this.
DS and DSD were playing an imaginary game. DSD was the queen and my DS asked to be the king she told him no absolutely not you have to be my servant.
With the butting in with adult conversations. Often me and DP will be discussing something in another room and DSD will be earwigging, come in and ask questions even though its a private conversation.
A few days ago, we were swimming and DP said to me its DSD world and we all just live in it. Which is true, it is. But no matter if I speak up or not DP wont have it. She will stick up for DSD or say I have a problem. Me and DSD have always got on well even in her previous testing times. I've praised her, supported her, watched her hobbies, cheered her on etc. But this behaviour on this holiday has been another level.
We went away 4 weeks ago for 3 nights and she was no where near this bad.
I know ultimately I have a DP problem because DP can also be rude to me at times. Get in moods with me and give me the silent treatment etc. So perhaps DSD is modelling this, I dont know.