Concerning Chrome causing issues with selecting in Fireworks, I noticed that it was the Google Music extension that was causing it. Once I disable the Google Music extension, the problem stopped and I can now have Chrome and Fireworks open at the same time.
If you don't have the Google Music plugin and Chrome is causing the issue, I'd be curious if there are any other plugins that cause this issue—maybe disable one-by-one to see which one is the culprit.