John, I believe there was a discussion of this some time ago; the only upshot
that I remember (perhaps because it was my own contribution, echoed by many
others) is that offering to copy and mail proposals is an excellent way to
encourage folks to include the grants/research office in the process.  Since we
have to get approvals anyway, offering a service to PI's/PD/s (i.e. copying and
mailing) tends to ease what they see as a burden (i.e. running their proposals
through our offices at all).
In answer to your specific questions:

1. Our office takes that responsibility, though some PI's/PD's prefer to do it
themselves--that's fine with us, as long as they get their final approvals
through us.  In that case, I obtain the signatures and then return the
proposals to them, exacting their assurance that it will go out as signed.

2. No one else requires us to do this, if that's what you mean.  We offer this
as a service to faculty, though we see it as to our advantage as well.

3. I tell folks that we need at least 24 hours to get approvals, copy and mail.
 Of course some folks give us less.  In those cases we do whatever we can.  If
we can't get it in the mail on time, it becomes their responsibility to drive
it to the post office at the nearest airport (35 miles away).

4. Fewer than 100.  UW-River Falls enrolls about 5,000 students with a faculty
of about 300.

5. We have a full time director (me), a 40% coordinator, and access to some
clerical assistance.

Regards, Bill Campbell
Director, Grants & Research
University of Wisconsin-River Falls