Dance schools are very often odd/competitive vibes as pp have said.
I’m sure you are a really kind person and maybe that’s part of the reason you find the environment a bit toxic. Did you do these kinds of activities when you were growing up? I’m guessing probably not or you’d be better mentally prepared for this kind of nonsense.
I am sure you mean well, but giving birthday cards to children in an outside of school class with your daughter at this age is quite unusual I think - especially if she is not close friends with them and you are not friendly with the mum. I would ask your daughter each time if she wants to give a card and only do it for the kids she actually likes. The best way to make friends is often to just choose individuals you like and build a relationship there, not just try and be friends with everyone.
As others have said, I think you need to try and let all this go a bit. Harking back to the age your daughter was when she started vs the age the other kids were - honestly who cares?! The other mums have obviously developed friendships outside dance (or are there all the time). You mention you wait for your other daughter to go in, perhaps they don’t all have multiple kids in tow and have had more opportunity to chat over the years, perhaps their kids are at the same school, perhaps they have shared values (eg. splashing money on costumes etc). That’s their look out.
Model for your daughter what not caring looks like - fake it til you make it if needs be. Take a book with you and get on with your own stuff. Don’t pander to them by giving unwanted cards etc (unless every child in the class does the same I suppose - how many has your daughter been given?) It’s important she learns to invest her care in those who also care for her, especially entering her teenage years.
Be perfectly polite with other mums, but completely distant, by choice. Let your daughter know she can do classes elsewhere and maybe she will gravitate to that after a while. Maybe dial down the number of competitions etc.
Once she starts secondary school she will have a lot of other things going on anyway and it’s great that she has other interests outside this. Encourage that and hopefully she will feel able to let go of this in her own time. Same with your eldest daughter if the same things apply to her experience at this dance school.
Dance can get to be a bit of an unhealthy environment for teens, so unless they want to pursue it professionally once or twice a week is probably more than enough…