You're perfectly at liberty to relicense public domain code if you wish.
The only thing you can't do is enforce the new license against people who obtain the code independently - either from the same source you did, or from a different source that doesn't carry your license.
If I use public domain code in a project under a license, the whole work remains under the license, but not the public domain code.
I'm not sure what the hullabaloo is about.
No, because they've independently obtained it from the same source that you did, so their copy is "upstream" of your imposing of a new license.
Realistically, adding a license to public domain work is only really meaningful when you've used it as a starting point for something else, and want to apply your license to the derivative work.