I personally don't think it would make a difference one way or the other, as long as the proper 301 redirects are put into place and all traffic to the old URLs is sent to the correct new URL.

I do, however, think the sub-directory approach looks cleaner.  Anything beyond the first sub-domain looks spammy to me (or just sloppy).  That's just me though.

Yes the presence of BingSiteAuth.xml or the lack-thereof doesn't really give you a definitive answer, but it does give you a little insight into the situation.

There is really no way to know for sure if they have activated or are using Webmaster Tools accounts unfortunately, unless you have access to their email accounts.

To 301 redirect all files in the direct to the new directory in .htaccess, try this:

RewriteEngine On
RewriteRule ^old\ foldr/(.*) /newfolder/$1 [R=301,NC,L]

This should redirect all HTML pages, images, you name it.  It should also account for the space too.

You'll want to move all your files to the new folder location so when the redirect happens your end user doesn't receive a 404 error.