Google - Chromecast Streaming Media Player - Charcoal

YES It gives you two options, fortunately: Plug both ends of chromecast into your tv.... one hdmi and one USB port necessary on tv to do this or Plug the hdmi side into an hdmi port on the tv and use the removable power cord to connect to a power source .

Yes defintely

you need a wifi connection

You would want to connect the hotspot to your Chromecast first. Once that is done then you would mirror your phone to the Chromecast through your Google home app on your phone and then it would show the Amazon or anything else on your phone to your TV.

As far controlling devices? Yes, as long as your computer or tablet is compatible(it would have to be really ancient for it not work) and are using compatible app. I personally use videostream for movies on my PC because it comes with a mobile app, meaning I watch on my TV movies that are saved on my pc, using my phone(there are probably newer/better apps out now, but I've never felt the need to look). Chromecast can also mirror a device's screen so it would show a PowerPoint presentation on a TV if it is also on the computer screen that is being mirrored. You would still have to use your computer to change slides, though.