1. I had the same situation, I told the toddler to go ask Mommy to swing. I'm not about to pick up or push an unknown kid in a swing. People are way too sue happy in this country, what if she fell? It sucks, but it's true.
2. You walk the playground and climb it with your child. Give the helper a big friendly "Thank you, but I'm gonna help him/her play today"
3. I would ask other parents kinda loud... is this your baby? Anyone know who's baby this is??? Then if there's no response after intense questions and thorough looking, wipe nose, call cops. Comfort kid.
4. Remove bully from on top of your kid, say "That's not very nice! Can you apologize?" Then bring it to the attention of their parents if it happens once more. If nothing else, remove your kid from the situation. Don't ever tell your child to hit another one (at this age... I suppose older kids may have to hit for their own protection one day)
5. Suspicious. Give her a diaper and some wipes, maybe some kid snacks and walk away. Strangers in your house or even car can seem a little risky. Then again, it depends on the woman and kids. To me this is all about the "vibe" I get from her. Do I feel she is just having the shittiest day ever and she is sucking at being a mom today, but is otherwise a potential friend and great mom? I'll take her home. Does she strike me as someone who does this often and may be preying upon helpers? Ignore it.
These are all responses from my "gut". I have no idea what's right or wrong according to playground rules. I make shit up as I go. The only one I am unsure of is the toddler wanting a push in the swing or up to swing. I love kids and would love to just help them out, but I would hate to get attacked by some parent freaking out that I'm touching their kid. UGH, the playground drama!