1.Can you list names and contact information of reputable banks which
have no US bank branches?
2. Which banks will allow you to open a corporate account online
without appearing in person, with the provision of all requisite
papers?  I need names and contact information.  Even better, I need
access to the online application (if any) to open the CORPORATE bank
account.
3. Of banks that satisfy both criteria 1. and 2. above, which ones
do not need to know corporate structure and beneficial 
ownership?
4. Of those in 3., which ones have the best internet based operational
features? 5. Cost of services is not a huge concern.
Thank you.

There are few if any "reputable" banks that do not have a presence in
the USA. Most important, ie reputable, banks derive a major portion of
their business from dealings with the US.

There is no way to open a corporate account in a reputable non-US bank
without a personal appearance before a bank officer who will require
ID and signatures.

If you want to open a corporate account, you will have to provide
detailed information on the corporation, its officers, its by-laws or
charter, signatures, photocopies, etc.

On the other hand there are hundreds of banks who will probably do
what you need. But are they reputable? Do you want to stick your money
in them?

You ask an improbable question since most banks which are dealing in
offshore locations without any presence in the US aren't quite
reputable. You might want to look in Turkey, Qatar or Hong Kong for
such "offshore-type" banks but if you dig a little, you'll see that in
one way or another, they deal with the US... a lot more than you
think.

So this brings us back to your initial need.

In my view, you're surest bet would probably be with a Canadian bank
account since the trip to meet with a bank exec (you might save
yourself the trip in exchange for some extra paperwork) won't cost you
too much, the funds will be very well protected and the online
services are first class, all the way.

After trying several corporate online banking solutions, this one
appears superior in both form and function:

http://www.tdcanadatrust.com/ebanking/easyweb.jsp

I hope this may help you in your quest.