After I embed flv video to Flash CS6 timeline, I only see black square and no video.
I go to
Import Video - mark option "Embed FLV in SWF and play in timeline"
Symbol type - Embedded video, all options marked except audio
Hit Enter or from Control menu - Test movie or Test Scene
You don't need a FLVPlayback component when importing it to the timeline, it creates a Video object and uses the timeline for playback.
Conrad~ You have a warning on your last compile (visible by the yellow triangle in SWF history in your screenshot). It appears to be a FLV but it either incorrectly marked an alpha channel or the codec used isn't wildly supported. I've really only seen this on transparency issues (because keyframes weren't properly marked).
What software are you using to produce the FLV and what codec and settings are you using for export? Did you use transparency?
Unfortunately I don't think we can recommend something like that as content on there tends to be copyright to its owner. You'd need to ask permission before grabbing the content.
If you have permission then I'd use Firefox with Flash Video Downloader but I download in MP4 format and then use Sorenson Squeeze (or Adobe Media Encoder) to convert that to an On2VP6 FLV. I've done it in the past for clients who lost their own content but had some up on YouTube. Use addons at your own risk and be sure to research them.