How to do an HD GIF?

GIF compression isn't very efficient for large animations; loading time and frame rate are highly dependent on the CPU and RAM available on the device. Since your questions 1, 2, and 3 all ask about significantly increasing the frame rate, frame size, and number of colors, be aware that doing so could make the animation load more slowly and run at a lower, more jittery frame rate for web site visitors with less powerful computers or phones, or if they are on a slow network connection. Almost all desktop and mobile hardware today has built-in optimizations for 30 fps playback of HD resolution H.264 video with low impact on CPU and battery, but GIF animation can't take advantage of that.

You can go ahead and continue exploring how to do what you want this, but you'll definitely need to test performance on the range of devices your client expects to look good on. You want to make sure that whatever you do isn't so large and heavy that it doesn't load or play as smoothly as expected, and you especially don't want it to delay the load time of the rest of the web page since that only turns visitors away.

One thing that makes the Headscape example work is that they seem to take care to animate only two very small areas of the frame. If they had animated large areas of the frame, the frame-to-frame processing load would be much higher, as well as the file size. Keeping the number of colors down was probably another conscious performance decision.

3) GIF has no resolution at all, so this is not meaningful, you can't increase or decrease the ppi because there is nowhere in a GIF to store a ppi. It is limited to 256 colours by design.

You could do more colors with an MP4 video the file size would most likely be close or smaller in size to the animated GIF which only supports 256 mapped colors.  You would not need that many frames and you would not be using a high frame rate.

The banner you pointed at is an animated GIF it has 73 Frames the frame size is 1300px by 472Pisels and is 1.3MB in size save as a mp4 video its size is 0.5MB.

I don't think that's a GIF, although there's no reason why you couldn't make one that size.  My connection is a middling 45Mb/s here, but it still took about 30 seconds for the animation to fully download and become fluid.

You are limited to 256 colours with a GIF, but it can be banner size.  1920 x 1280 and 256 colours with a bunch of frames would run to several megabytes, so you'd need to think about that.

I'd be inclined to ask the same question on the Dreamweaver forum where they would know how to build that sort of graphic with Animate and run it in HTML5 (at a guess — I don't do websites)