Well the iPad might work for you. It can all you need it to except homework. However you can get iWork for it which depending on what you have to do could work. The iPad is at the top range of your budget and you will have to wait 60 days to get it
I don’t think you can watch DVDs on it.
No but you can play itunes, so any movies or TV shows you have in your account can be played. And a lot of Trek series are actually a LOT CHEAPER on itunes than on DVD.
And it uses Quicktime in place of Flash or Windows Media Player, so streaming might be an issue too.