A: Yes, but only if you play the content on an iPhone or iPad. Open the Amazon Prime video app on your iPhone or iPad and play whatever movie or TV show you want to play. Click on the AirPlay icon in the upper right corner of your device, and when the menu pops up click on Apple TV. The content will now stream through your device to the Apple TV and play on your television. Your iPhone or iPad must be connected to WiFi for this to work.
A: Any HDMI cable will work. Expensive HDMI cables generally don't offer any better performance than cheap ones. Since digital signal either gets through or it doesn't, there's no real reason to worry about degradation or interference. A $3 cable works just as well as a $60 cable. The only reason to pay more is to get something more durable if you plan to be moving devices around a lot or plugging and unplugging stuff. If you just plug it in and leave it, you should not have any problems with the cheap cables.
A: Though there's no "Computer" icon in any of Apple's marketing materials for the new Apple TV, the company's website confirms that Home Sharing will remain a feature in tvOS: Under the System Requirements section of the box's tech specs, Apple notes that "iTunes purchasing and renting and Home Sharing require iTunes Store account." Not the most graceful way of acknowledging Home Sharing's continued presence on the Apple TV, but it's there.
A: Absolutely: AirPlay is enabled with the new Apple TV, as is full-screen Mirroring. Even better, the set-top box supports peer-to-peer AirPlay, so your friends never have to connect to a Wi-Fi network to use AirPlay.
A: If you're TV has a spare HDMI port your right to go. Even if you don't have an HDMI port you can get a converter cable. I run one Apple TV with a converter cable to a 2004 model Fujitsu plasma and it runs with no issues.