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As best I can tell:
Adobe Animate was formally called Flash Professional. About 10% of apps for for the app stores are make using Adobe AIR, and a lot of those are created in Adobe Animate. It also continues to be used for web activities and games, as well as for banner and video support for browsers that can't manage HTML5. And it's used by animators for creating TV shows and general 2D character animation.
Character Animator is an After Effects add-on, that can track your facial movements in order to animate a virtual character, using art made in Photoshop and Illustrator.
Fuse, formally known as Mixamo, is for creating 3D characters. Those can be exported for other uses, but the Adobe use for them has been to create posable virtual characters as placeholders in Photoshop brochures. The implication is that you would swap for a picture of a real person later on, after the general design was approved, but you wouldn't have to! Currently Fuse is not being worked on, the same team is working on Felix.
Project Felix is for creating 3D scenes, with template models and primitives. This video might give you an idea about it: First Look at Project Felix | Adobe Creative Cloud - YouTube
3D layers in Photoshop have been around for a while, and could be used for either Fuse characters or Felix scenes, or just imported Collada models.
You didn't ask about it, but Edge Animate is an HTML5 CSS animation tool. Its features overlap somewhat with Adobe Animate, but it only does a small amount of what Adobe Animate can do. But the things it does are done differently, and is more suitable for say long scrolling web pages. Edge Animate is not longer developed, but you can still download it. If you're interested in that sort of responsive web page creating you should also look at Muse.
So, the short answer is that all of the things you asked about are for different things, with some small overlap, and a bit of confusion when it comes to having similar names!
Thank you so much - I do really appreciate the help. It is so difficult to know where to start really. I'm so interested in making scenes move then putting still photos of people in them and animating them too. I'm enthralled! Don't know where I should really start but thanks much.
If you want to watch an expert animator working in Adobe Animate, look at Twitch TV at 1pm Pacific time today:
Mike Milo does a three hour stream where we all watch him creating something in Animate. There's live chat too, so you can ask Mike or the other viewers questions about Animate, or about what Mike is showing.
Thanks for the kind words, Colin!
Carol, I think that Adobe Animate would probably suit your needs the best as many little films have been done using photos as the heads. Jibjab.com made a whole career doing just that very thing and they used Flash to do it which is now renamed Animate.