The only thing I can say about why Emma so readily agreed to help Sidney is that Emma was so blinded by getting back at Regina that she didn’t see through his lies. (But she wasn’t the only one fooled, everyone in on that counsel meeting immediately approved once Regina revealed that she was building a playground for the kids. What a wishy-washy bunch!)
You make a good point about the building of the playground. It makes no sense that it would be built that quickly and bypassing all safety procedures…Emma should have had her arrested for that. Not to mention at the beginning of the episode, Regina chastised her for allowing Henry to play at the playground castle while it was in disrepair because of the storm.
It is very curious that Regina would need that much land just to build a tiny playground out in the woods, of all places.