Can I watch a movie I downloaded via Apple TV on my iPad?

If you purchase it from the iTunes store the you will be ablle to go in to the itunes store on your iPad, look at ourchase and download to there.

Or you could go the long way round, Download to your computer and sync it throught to your iPad

I have not tried rental films but in theory yes. Give it a go.

No, unfortunately it's not the same for rentals. Rentals can only be viewed on the device they are rented on, with the exception of a computer. So if you want to view a rental on multiple devices, rent it on a computer.