I would have put the building numbers as ref=* and 'Department of Whatever' as name=Department of Whatever (or name:en=Department of Whatever)

You probably have wireframe mode on. Ctrl/Cmd-W (I think) or use the View menu(?).

I'd use "fixme=Incompletely surveyed" or something to identify a way as being …incompletely surveyed.

I've been using the WMS layer in JOSM for just under a week now and I've got quite a lot of tracing done; no emergence of this alignment shifting, though.

Are cyclists welcome on the path/track?... or is it a case of "there's no ban on cycling"? If cyclists are welcome, I'd suggest highway=cycleway, surface=surfaced (or =tarmac), bicycle=yes, foot=yes, width=[whatever the width is], etc. If cyclists aren't *signed* as welcome, I'd suggest highway=track, surface=surfaced (or =tarmac), bicycle=unknown, foot=yes, width=[whatever].