I agree that modelling is one of the best ways to teach about desired behavior(really anything). Honestly, the examples you gave don't strike me as controlling so much. Just sort of typical for young ones. Sometimes I get a lot of resistance to whatever it is I'm trying to get us to do
and many times it's because I'm trying to go too fast without giving ds any idea what to expect. I've noticed he likes to be warned about upcoming things. So maybe with your swim class example, on the way to the class I might remind her of the routine and what's going to happen when you get there. Might look like this:
You: when we get to the Y, Daddy will drop us off, then he'll park and come in.
Her: no!! Want daddy to come in too!!
You: oh, you'd like to go in at the sane time as daddy?
Her: yes
(at this point you know what she wants and can decide if that will work or not)if you decide that won't work, then
You: it would be fun if we could all go in together, but we're running late so daddy's going to drop us off and come in after he parks the car. Please get ready for that.
For us that is usually enough to diffuse any resistance, but if she's really insistent, what would happen if you agreed to do it that way- all going in together? Seems like if that would satisfy her, it might be better than harshly taking her out of the carseat and her having a fit in public. Not judging you, just trying to help.
I try to say yes as often as I can and give ds as much control over how he does things as possible. I also try not to get into power struggles with him, he doesn't have anything else to do, but I do! YKWIM?
Anyway, you said your dd's behavior brings up issues with your mom, have you read any of Naomi Aldort? I really liked her book, Raising Our Children, Raising Ourselves. Very thought provoking.
And about the wanting help to find the doll's shoes, honestly, I wouldn't make her say the whole sentence when you knew exactly what she wanted. I'd just say, oh you'd like for me to help you find your doll's shoes? I'd be happy to. Let's look for them. Where do you think they might be?
That may seem to some like spoiling or giving in, but it has really worked well for me. It's not what I would have thought I'd be doing before I had my own child, mind you! I thought it was best to be super strict, but it feels like I'm doing the right thing by respecting him as a person and trying to be as generous with him as possible. He's learning how to treat me and others in every interaction I have with him. Again, to me, it goes back to modelling. I also found Unconditional Parenting to be a real thought/approach changer.
Good luck!