She reminds me so much of my son. Not the temper trantums, but the way nothing affects her behavior. My dh has gone to get him and put him in time-out and I swear he laughs. He also finds ways to have fun in time-out. There are times I just don't know how to get through to him. Lucky for us he's the easier child and rarely gets in trouble.
One thing that works great for him is positive reinforcement. Since your child is a very positive child (not letting anything get to her, lol) she might listen to that much better. For example, when she puts a toy away, praise her for it. Try to focus on the good things and making her feel really good for making those decisions.
With temper tantrums, they are a way to get attention. It's a way for her to get what she wants. I've been told over and over to just simply walk away when she does that. Pretty soon they realize they're not getting attention and they stop. Of course in public you can't just walk away. But, try totally ignoring her. I've seen kids go into screaming fits at stores and the mom just ignores them and keeps shopping. It cracks me up, but they're doing the right thing.
I found 18 months to be a hard time to discipline. I was never sure what my kids did and didn't understand at that age. That's why focusing on the positive part seems to help.
Out in public, my ds wouldn't hesitate to run off. My dd is older and when she was younger, she would never, ever walk away from me. My ds is more independent and would be gone in a heartbeat. So it's just no choice for him but to go in the shopping cart or stroller at that age. I worried too much and wasn't about to take a chance. Sometimes he'd get mad, but I just ignored it and he finally realized that he was being strapped in no matter what. Sometimes I'd let him walk beside me but as soon as he took off, in he went. It didn't take long for him to get the picture. And, all those quiet kids you see out in public have had their moments too. So I'm sure each mom can relate to what you're going through.
So I'd try focusing on the good things she does during the day. When she gets a lot of attention for that, she'll want to continue that.