If I have a Perplexing chimera out and my opponent casts a spell that targets something, will using strionic resonator to copy chimera's ability exchange control of chimera back to me while still allowing me to choose the new target for the spell?
Yes. It’s a lovely evil combo, as is explained by one of Perplexing Chimera’s Gatherer notes:
2/1/2014: In some unusual cases, you may not control Perplexing Chimera when its triggered ability resolves (perhaps because the triggered ability triggered again and resolved while the original ability was on the stack). In these cases, you can exchange control of Perplexing Chimera and the spell that causes the ability to trigger, even if you control neither of them. If you do, you’ll be able to change targets of the spell, not the spell’s new controller.