Yes, I know, most of you have probably left floppies behind in the dust, but they're still handy for some small quickie things.

Here's the question. Floppy disks have a limited lifespan, of course (though I suppose everything does), and eventually at some point as it starts to decay, it's normal to get a message something like "Failure reading A-drive." But twice recently I've had messages of "Disk not formatted. Re-format?" That's when the disk is not only formatted but is carrying some files.

Should I assume the problem is the floppy, or could there be a problem with the A-drive where the file is being copied? This is a newish (for me) hand-me-down computer, which is likely about a half-dozen years old, although I've been using it roughly a year or so.

You got it right both times. The floppy drive and the disks are old and the reader or the disk are not 100%. Try getting a needle or similar and trying to turn the edge of the actual disk inside the holder (not the drive) some time they get a bit jammed up with dirt and will not spin.
If you get it working, give Noah a call he might want it back. lol