Downloads no longer appear on the bottom of my browser window, or in a cascade from the dock.

If you can find the downloaded file and highlight it, but can't "use" as it's intended, you have more problems than simply an inability to locate an icon.

"There is no application set to open the document "name of document". This means you have no app installed to use the file as intended. For example, if you download a .exe file you can click on it but it will not run. It's a windows file. Or it could mean you have the app but you've not set it as the proper default app.

The same will be true for any file of any type ... without the correct app set to use it, nothing will happen other than an error message about not being able to use it.

What specifically, by name, is an example of a file you cannot "run"?

I agree, this is easy. But something has changed on your system and you've not mentioned other users.

One way to deal with this is to right click the file and choose "Open with". You'll then be presented with apps that may run or use the file. This method will also show you what apps your Mac thinks is there to use the file ... if any. It will also indicate the default app if there is one.

Have you looked in your Downloads folder? That's where they should go unless the path was altered. You can access them from there, or whatever the new folder they may be in.

In Safari they are at the top in this button:

and have ben up there for as long as I can remember (in Safari)/

"I don't have that in safari, and the usual download mechanisms aren't in Chrome either. Did I somehow disallow downloads?"

It should appear in Safari when you start the download.