Phil,
No - Adobe does not allow concurrent licensing of it's products. If you have 20 folks that need to use it Adobe will ask that you purchase twenty licenses. Not any different than MS-Office or other application.
However, at the 20-seat rate you should contact a reseller such as CDW, Dell ASAP, etc. and see what sort of licensing discount you can get. It may not be much, but $5 saved is still $5 you didn't have to spend.
As for their Creative Cloud system - same thing applies - individual ID's, accounts, etc.
Finally - can I put Acrobat on a server and have it monitor a watched directory for work - no. Explicitly prohibited in the EULA.
Doug