I want to add a YouTube search box to my Web site. Can it be done?

Hi Dave. I wanted to show my 12th grade a documentary about Fermat's Last Theorem. I was planning to send them the link via e-mail, and have them watch it at home. But the link (below) has a porn site listed under "related videos", so I don't feel that I can send it to them.

  http://video.google.com/videoplay?docid=8269328330690408516

Do you know of a way for me to download that video onto my computer?

I'm addicted. First it was America's Funniest Home Videos, now it's YouTube. There are just too darn many hilarious videos on the site. What I really want, though, is to be able to add a YouTube search to my own site so people can easily find all the funny stuff there. Is that doable?

I just got involved in a charity event we're planning for later in 2008 and one component of it is going to be a series of videos we release through YouTube. Is there some way that people can find out about them without us having to send out a never-ending series of press releases or updates?

Is there a way I can use Web phenomena such as MySpace and YouTube to get more clicks on my affiliate site?

I've been blogging for a while now and am having a good time, but I'm now a bit puzzled about how to expand into the next wave of content: video. Specifically, I bump into funny or informative videos on Google Video but have no idea whether I can embed it on my own page and if I can, how to do that. Can you offer some pointers?