No, you do not have to have an Apple TV to connect your iPad to the TV. The Apple TV adds convenience because you can display video from your iPad wirelessly on your TV by using the Apple TV. You can connect your iPad directly to your TV but you will need to buy an AV cable.
I believe you can connect your iPad to a suitable TV and view the film on your TV screen. I do not believe you can download the iPad film onto your TV.
Yes you can purchase an HDMI, VGA and composite video adapters to connect an iPad to a projector. You can also use Airplay Mirroring to Mirror the iPad's Image on a TV or projector using AppleTV.
You don't use the Apple USB cable. This is for syncing with a computer. You need to purchase an iPad HDMI or VGA adapter, and connect it with those cables.
You have to purchase a separate adaptor for this. You can do av output, hdmi output and even connect it with AirPlay onto your Apple TV.
You can use Airplay with an iPad and and AppleTV to push video from the iPad's Video app, Netflix and other apps to your TV.
Put your iPad on the router.
No. You will need to connect the iPad to a computer with iTunes to sync.
The iPad can output either by cable or WiFi. It cannot receive from the TV.
Yes it will.
Time flies and you can mirror your iPad to Samsung TV via Airplay now. Just follow the steps below. Take your iPad, unlock it and open the video that you want to watch on your Samsung TV. You'll then see the Airplay icon, click on it and the list of Airplay devices will appear. Select Samsung TV from the list of Airplay devices. Play the video by streaming it from iPad to Samsung TV. Make sure both devices are connected to the same internet. If you want to stop the video, click on the top of the screen and select the play button, so the already playing video will stop.
Yes, connect your iPad to Itunes and click restore.