I'm using Vegas Pro 12.
The .flv file cannot be dropped on the timeline. So I converted it to .avi, doesn't work either. Then I converted the .flv file to .mp4, was able to drop it on the timeline. But still can't play it. When I tried to play, I see the marker moving along, but the video in the Preview window is frozen on a previously played part of the timeline. What am I doing wrong?
Thanks,

flv is not among the supported file formats of Vegas Pro:
Vegas Pro 12 Technical Specifications (http://www.sonycreativesoftware.com/vegaspro/techspec)
You can demux flv FLV Extract 1.6.3/2.1 jofori89 - VideoHelp.com Downloads (http://www.videohelp.com/tools/FLV-Extract)
into elementary streams and then mux them in mpeg transport stream
tsMuxeR 1.10.6 - VideoHelp.com Downloads (http://www.videohelp.com/tools/tsMuxeR)
Or You can download a mp4 file from youtube.

As Juris noted, flv is not supported, period.

AVI is just a container format - there are numerous codecs that can be used within an AVI.

So, at a guess, what you are doing wrong is using some consumer video converter with its default AVI output settings, whatever those are. What converter are you using? Wondershare, for example, likes to do what it does, with codecs, sizes & etc. not easily controlled, a bad choice for later pro editing.

What codec choices do you have with whatever you're using to convert to AVI?

Likewise, it seems a similar story with the .mp4 conversion...

Go to Youtube, copy the URL as the vid is playing,
Go to KeepVid: Download and save any video from Youtube, Dailymotion, Metacafe, iFilm and more! (http://keepvid.com/)
Press Enter,
Download the mp4 version in the highest quality/size available,
Drop in Vegas,
Set Project to video parameters when asked,
Render to lagarith, dnxhd, cineform etc, whatever your intermediate of choice is.
Unless you are doing something very quick, I would transcode it, Vegas doesn't seem to like the youtube mp4 files, they have been pretty unstable for me.