Kindles are really hard to root. Use XDA Developers forum for this kind of thing. https://xdaforums.com/t/fire-hd-8-2018-only-unbrick-downgrade-unlock-root.3894256/, which is what that guide is based on, looks like it only works for the 2018 version and seems way more difficult and risky than most ROM installations. Also, that won't install Linux, just a different Android version.
LinuxSBC
Honestly, an actual ereader might work out better for you if that's all you plan to do.
You can't do that. Installing custom ROMs on Android devices is very different than installing an OS on a desktop/laptop. Most devices don't allow changing the OS at all, most of the ones that do don't have any Linux builds, and then you'll be stuck with whatever distro you're given (probably UBPorts or maybe PostmarketOS) rather than choosing your own.
What do you think an API is? They have reverse engineered the iMessage API and are using that to connect to the iMessage servers. It is literally impossible to do as you suggest (use entirely their own resources) because iMessage is centralized and cannot federate with any other server, even if one did exist.
How? It's not a MitM or anything like that, it's connecting exactly how an Apple device would connect. Everything is still E2EE, just one of the ends can now be an Android device.
I think their RSS feed has a placeholder title for this.
Vivaldi does too. It's nice.
From what I understand, their guess is that Apple is now checking if the device also has support for other services, such as FaceTime. Beeper Mini and pypush don't pretend to support FaceTime, so it breaks.
Their hope was that they got close enough to an actual Apple device that breaking it would break Apple devices. It turns out they weren't close enough, but they could be with a few improvements.
Because they can't break that. It's using real Macs, so if they break iMessage for Beeper Cloud, they break it for their customers.
Also, KOReader adds a bunch of extra features and functionality.