The take-away: your mum didn’t want you to pay because she’s lovely and loves you and wants you to keep your money for yourself. I understand that you wanted to treat them, and sometimes it feels like you have to fight them to see you as an independent adult and let you treat them. She’s not trying to isolate you, she’s trying to mother you.
The movie: they were fed up of waiting for you, and, trying not to hurt your feelings, made up an excuse for starting without you. They were not trying to isolate you, they didn’t want to wait for you.
You are being over-sensitive, but they’re being a little under-sensitive. That’s families, I’m afraid.
My advice is, if it’s important to you, be assertive. For example, with the take-away: “I really want to treat you all. You cooked last night and I want to pay tonight. Let me treat you.”
Sulking upstairs just makes you look like a moody adolescent. If you want them to start treating you like an equal adult, you need to behave like one.