We are thinking about printing our documents in-house. We need to
identify the types of machines/equipment that would, in high volume,
reliably do the
  - stapling (center of page, 7 inches from edge, preferably both
staples at the same time, electric not manual)
  - folding (from 14 inches flat to 7 inches folded, cover stock
[heavy paper])
  - cutting (large stacks of paper, to the right size .. not just a
few sheets)

So for each of these three pieces of equipment we need:
  - The name that it is called in the industry 
  - Preferably two or three brands/URLs to purchase
  - General price ranges

john9372...

In the industry, all the equipment you mentioned
comes under the heading of "bindery equipment".
Staplers are called staplers and stitchers, 
and the common method for stapling
the middle of a page is called "saddle-stitching".

Here's a page where you can buy a new
Salco R106 Professional Electronic Flat
Saddle Stapler - $512
http://machinerunner.com/Electric-Binder-Staplers/Salco-R106.html

That one uses a single stapler-head, and the 
pages are moved through manually. This one
has dual stapler-heads, and can accomplish
double-stapling at once.
Salco R106 Gangster Professional Electronic
Flat Saddle Stapler - $1,021
http://machinerunner.com/Electric-Binder-Staplers/Salco-R106-Gangster.html

Stitchers are machines which use a spool of wire
rather than a magazine of staples. Here is a page
with a single and a dual-head stitcher, from 
$1,930 - $3,700:
http://machinerunner.com/Binding/Stitchers/Wire-Stitchers.html


Folders vary in their ablility to handle different weights
of paper. You specified "cover stock" which begins at about
50# paper weight.

The "friction" folders on this page range from $999
to $7,583 for machines which will handle 50# weight
or higher:
http://machinerunner.com/Paper-Handling/Folders/Friction-Folders.html

The "vacuum" folders on this page, range from $3,667
to $14,394. These utilize vacuum cups to grip the 
paper, and can handle much thicker stock, up to 110#:
http://machinerunner.com/Paper-Handling/Folders/Vacuum-Folders.html


Paper Cutting Machines, or cutters
range from manual (probably not what you want)
to semi-automatic, automatic and programmable.

Manual cutters range from $489 - $1,938, with 
sheet capacities from 200-700 sheets:
http://machinerunner.com/Paper-Handling/Paper-Cutting/Stack-Cutter-Paper-Cutters.html

Semi-automatic electric machines range from
$2,713 - $6,667 on this page:
http://machinerunner.com/Paper-Handling/Paper-Cutting/Semi-Auto-Electric-Paper-Cutters.html

Automatic electric machines range from
$10,802 - $17,861 on this page:
http://machinerunner.com/Paper-Handling/Paper-Cutting/Automatic-Electric-Paper-Cutters.html

Programmable machines range from $5,151 - $25,571
on this page:
http://machinerunner.com/Paper-Handling/Paper-Cutting/Programmable-Electric-Paper-Cutters.html

As you can see, many brands are represented
in each category of machinery named above.


The above represent the range of prices for 
new machinery. Of course you can obtain used
equipment for a smaller investment, however
used bindery equipment sales are much like
used car sales, in that specifications may
not be as detailed, and it is much easier if
you are already familiar with the capabilities
of a particular machine you are looking for
by name.

Below are some sites which feature used
bindery machines:

GE Capital:
http://geasset.adbsys.com/Application/Storefronts/EventCategory.asp?CategoryId=8A5A12A3-C723-4352-8099-BAFB61539BF8&VendorId=E253FAC2%2D1E27%2D4C10%2D9E7C%2DA21D9F9B9C5F&EventId=6782166A%2DE96D%2D4C78%2DA742%2D7DB3DFEE91E1

EBAY:
http://listings.ebay.com/aw/plistings/list/all/category26240/index.html

PrintTrade.com:
http://www.printrade.com/

CFS:
http://www.copyfinishing.com/used1.htm

Used Printing Machinery UK - worldwide trade:
http://www.used-printing-machinery.co.uk/


Searches done, via Google:

"bindery equipment"
://www.google.com/search?num=50&hl=en&lr=&ie=UTF-8&oe=UTF-8&q=%22bindery+equipment%22

"bindery equipment" on Google Directory
://www.google.com/search?num=50&hl=en&lr=&ie=UTF-8&oe=UTF-8&q=%22bindery+equipment%22&sa=N&tab=wd&cat=gwd%2FTop

"new bindery equipment" price
://www.google.com/search?num=50&hl=en&lr=&ie=UTF-8&oe=UTF-8&q=%22new+bindery+equipment%22+price

"used bindery equipment" price
://www.google.com/search?num=50&hl=en&lr=&ie=UTF-8&oe=UTF-8&q=%22used+bindery+equipment%22+price

If you need further assistance before rating this answer,
please feel free to post a request for clarification.

sublime1-ga

Wow. An excellent, comprehensive, and extremely helpful answer. Thank you.