Thanks for sharing your experience, and the reminder about how to use the site.
I've use SnippingTool many times in the past, lots of diff apps, without any problems.
After doing some experimenting, the problem seems to be the browser, ie. when pasting directly from SnippingTool into Internet Explorer 11.42 (yes, I know it's old tech, but still good for some things), and FireFox 45.0, both seem to do something, because my screen flickers momentarily, and then the "saved" status appears (bottom right of editor), but nothing appears in the editor, the preview or the saved post.
@phase21 It's difficult for me to advice anything to you, because I can clearly see that you are an advanced user - definitely not terrified when asked to do a screenshot but maybe try using another program (for instance this one - free for personal use) or choose another format - I recomment *.PNG extention.
BTW instead of pasting directly, you can check if a traditional way of adding an image (save/upload), works for you:
... but if it's a matter of incompatibility between outdated browsers and the latest version of Discourse, it probably won't help ...
Why are you still using Firefox 45 while even on old/ancient computers you can have Firefox 52.9 ESR?
The pic I included above was by using SnippingTool (Windows 10, Accessories), and pasting directly into the editor using MS Edge as the browser - those two work together Ok. So, I suspect that its older browsers that can't cope with Discourse's newer features.
I haven't tried Chrome yet, and will definitely check-out PicPick.
Using a screen capture tool is ideal for this sort of environment, where you simply want to paste the image directly into the editor, and don't want/ need to spend a lot of time on it. If a more detailed explanation is needed, then I usually save the image to disk, work on it, then upload.
In regard to Firefox, yeah, luddite, busy with other things, comfortable with "old house slippers" - usual excuses. I'll upgrade soon, I promise - change is good, it keeps us vital.
You need modern browser for the Discourse. If you use old browser some features required by Discourse may not be supported by the browser. This applies to Clipboard support in the browser https://caniuse.com/#feat=clipboard. As per this page, IE 11 supports only copy-pasting text and URLs.
In such cases you may need to use method described by @Milosz
As they say on Discourse pages, "IE is barely supported":
but maybe try using another program (for instance 
- usual excuses. I'll upgrade soon, I promise - change is good, it keeps us vital.