While I've never heard of that message before, Google has:

https://www.google.com/search?q=%22No+object+for+moniker%22&ie=utf-8&oe=utf-8

Some threads refer to Group Policy security restrictions, another mentioned desktop encryption software, and then there are a bunch of sites suggesting you download some fixit tool (be very suspicious).

What if you first create a folder on your desktop (e.g., FxDown folder). Will downloads save to that folder or does it inherit the same problem?

I was able to create a folder "FxDown" and choose it as my download folder (And thanks). Then I changed it to "Incoming Downloads" with the space, and that worked too.

Then, just for fun, I tried again to select Desktop, and got the

"Desktop no object moniker"

error message again.

It seems as though it must be a permission or policy issue, but the wording is so strange.