Also, don't get anything from Vizio or Olevia. They are cheaply price, but they are just that; cheap. I'd say anything from Samsung, LG, or Sharp would be your best bet.
Yes, you can do that. I've never run a PC on a high def TV, but many people do. From what I understand, gaming is better on a LCD PC monitor due to quicker response times and higher resolutions (more pixels). Plus, it's kinda hard to sit 3 feet from a 46" TV - you'll need to sit back 8-10 feet. However, a television will obviously give you more versatility if you like to watch movies and what not.