The film will need to be downloaded in order to watch it, you can't stream it from the store. Whether you can re-download a film will depend upon what country that you are in and whether it remains in the store (studios occasionally remove them). You are better off keeping your own backup copy of all of your downloads rather that relying on being able to re-download from the store in the future (you can copy purchases from your iPad to your computer's iTunes via File > Devices > Transfer Purchases)
What can be re-downloaded where : http://support.apple.com/kb/HT5085
You can also get wifi-enabled drives which allow you to stream your content to your iPad, so not everything needs to be stored on it e.g. GoFlex Satellite (there are other makes as well).
If I buy a movie on iTunes, can I download it onto my iPad and keep it there?
If I then delete it, can I download it again free?
As long as re-downloading of movies is available in the iTunes Store in your country and the movie hasn't been removed by the content owner, yes. It's best practice to keep a backup of all your content, though.
If I do not have my iPad with me, can I watch it online through iTunes?
You can re-download the movie to the computer, but movies are not available as true streams as, for instance, those from Netflix are.