The only toy we ever bring to the playground is bubbles, and I blow them. All the kids can chase them if they want, and it doesn't create tension. When it comes to other people's things, I just tell her "Honey, that's not yours. Let's go do XXXX." and then I move her to somewhere else. With repetition, they eventually get it. (Mine now tells me "Mommy, that's not mine" when we see other people's things...)
As for the tantrums, that is normal. My daughter is 2 and we still occasionally get them when it is time to go. I warn her several times in advance, and when it is time to go, we go. I put her over my shoulder or in the stroller or the carseat, and we leave. The tantrum NEVER gets more time. Over the last year, the amount of tantrums and the length of them have shortened considerably. Don't completely avoid the park for them! It gets better, and they can't learn if they don't experience the situation.