Kindle/Android Setup: M4B Version
Here are the steps I used to successfully play the M4B version of the audiobook on a Kindle Fire tablet (Fire OS 188.8.131.52). As far as I know, this same procedure will also work on an Android tablet or phone, but I had no way to test that to be sure. You may know of or find another method that works just as well or better. However you manage to accomplish it, the ultimate goal here is to get the M4B downloaded and played in the Listen Audiobook Player app.
Listen Audiobook Player is the best M4B player I was able to find for Android, in fact, it was the only one I could find that worked properly, allowing full use of the M4B's features, such as artwork and complete navigation of the audiobook's chapters. There are other players out there, but many of them are missing key features or are clunky and difficult to use. That includes the software that comes pre-installed with the Kindle (and presumably an Android). I also didn't have good results using VLC, a commonly used player that's normally good for just about anything, but it doesn't handle M4Bs very well. And while Listen Audiobook Player isn't free, it's only $1.99 and works really well.
However, there's a problem, at least if you're using a Kindle as opposed to an Android. Listen Audiobook Player is only available from the Google Play store, but Kindles are initially set up to only be able to purchase apps from Amazon. To get around this (and don't worry, this isn't illegal), you need to install the Google Play store on your Kindle. In order to do that, you'll first need a file manager app of some kind. The one I settled on is File Explorer for Kindle Fire, or you may prefer X-plore File Manager. Both are free and relatively easy to use.
Once you have one of those and are comfortable with it, carefully follow the instructions outlined on this page (an external link that will open in a new window):
How to Install the Google Play Store on an Amazon Fire Tablet
After that, you can install Listen Audiobook Player.
Unfortunately, unlike my more detailed guide for downloading and setting up the M4B in iOS, I can't provide screen captures in this one because something is wrong with my Kindle where it won't take screen caps. (It's supposed to, but it doesn't work for some reason.) Instead, I'll do my best to walk you through everything via text.
When you purchase the M4B from this site, you'll be directed to a page with instructions on where to download it. This leads you to the audiobook's download page on the file hosting site MediaFire. You'll see a blue rectangular download button stating the M4B's file name and file size; be sure to tap this and not one of the ads that MediaFire sometimes puts up that are made to look like download buttons but aren't.
Once the download is complete, the M4B will be in your Download folder, which you can confirm by navigating to it in your file manager app.
Next, open Listen Audiobook Player. If you're opening it for the first time, you'll need to set it up, which I found a little difficult. It asks you where to store audiobook files, and you have to first specify that you want to use internal storage, then pick a folder. I chose the Download folder since I knew that's where the M4B was; the file path is /storage/emulated/0/Download. Of course, you could also choose a different location, but then you'll need to move the downloaded M4B to there (using the file manager) in order for Listen to find it. This location be changed in Listen's settings, accessed from the small three vertical dots icon in the upper right of the screen: Settings>Library Folder(s). You can even have more than one location, though I never tested this myself.
Once you've opened the M4B in Listen's Library, it will start playing. The artwork also serves as the pause/play button. To get to the table of contents, tap the chapter name at the bottom of the screen. There is a scrubbing control (a horizontal line) above the artwork for searching through a chapter, and one thing I didn't like about the default settings is that this initially just shows the entire length of the book, not of each chapter you're in. This can be changed in Settings: Tap the Library icon in the upper right to get to the Library, then the small three vertical dots icon>Settings>Appearance>Show Chapter Time (and you can hide Show Book Time if you want). Back in the playback window, there are also arrow buttons at the bottom to skip forward and backward at either ten second or one minute intervals. These lengths can also be changed in Settings. Overall, Listen Audiobook Player works really well precisely because it is this customizable.
And that's pretty much it. Enjoy listening.