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.
A wireless keyboard is not needed with a Smart 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.
In order to connect your smart TV to a Wi-Fi hotspot you must have the password. If your connection fails, you can check to make sure your adapter is working properly.
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.
Check the owner's manual for instructions on pairing the headphones to the TV.
Yes, the new iPad still is usable with the Smart Cover.
Composite video input for video and analog audio cables for audio.
Yes there are versions of the Vizio TV that connect to the internet through smart TV functions often found in modern sets. They can cost anywhere from $300 to $1500 depending on the size.
You will need to select to view featured videos before getting to your YouTube playlist. You will sign in to your account and then find the videos you want.