Well, one thing I see that could be changed is your DirectX version. If you look at the bottom of the DirectX Diagnostic Tool image you posted here, it says you have DirectX 6.1a(4....). I suggest you download
DirectX 9.0c and install it.
Then, let us know if that helped any in playing YouTube videos. If not, or if it helped some but you still don't have smooth play, I believe you really need to install a video card to replace your current card or onboard video (which I believe is what you have).
You've said you have broadband Internet access and that your Internet speed "will stream pretty fast". Exactly what type of Internet service are you using? DSL or cable? And, what's the actual speed? If you don't know, go to dslreports.com (go here:
http://www.dslreports.com/tools) and run a speed test.