Is it possible for viewers to watch internet TV through my website like on a drop-down box? In other words, using software from my website, can viewers watch TV and also search on my website?
Do you mean live streaming real TV? Or simply youtube videos that you choose?
If it's just Youtube, it would certainly be possible - many sites play advertising to viewers while reading regular content. There is a wordpress/joomla plugin to just that here: http://www.dpopup.com/blog/?p=624 though you will need to a little coding yourself.
I'm not sure how useful the functionality would actually be though. If I wanted to watch TV while browsing the internet, I would open my own tab for youtube, or get my own tv-card. If the video is relevant to the content, it should just be added inline in the webpage.
So in short, yes it is possible, but please dont do it!