I'm also running Win7 X64.
It includes two versions of the IE8 web browser: 32 and 64 bits.
The Flash player can be installed, and will run, in the 32 bit browser. (Not so in the 64 bit browser.)
I believe that there was a problem a few months ago with the Adobe website refusing to install the Flash player on a 64 bit version of Windows. (I think it affected Vista as well.) There was a work-around: an installer could be downloaded and run locally. That is no longer necessary: my current Win7 X64 install was a clean installation, and I was able to install Flash auotmatically on the web. YouTube works fine (but only in the 32 bit browser).
I can't explain your problem, unless you have somehow managed to set things so the 64 bit...