I would like to open a Euro-denominated account (checking and/or money market)
in a US bank (with a USA address and USA branches).

I would like to have this because my income comes from
Europe in Euros and I am planning to spend it in Europe, so I would
like to avoid paying double change fees (EUR->USD->EUR) and keep the money
in EUR all the way.

It can be either personal or business type of account.

Dear Aderminder,

The best way to keep Euro in Euro is by opening an account with a
Euro-land bank, whereby an online banking facility would probably be
most convenient for you.
There are many of these in Europe.  Some may not be so receptive to US
applicants, and I cannot say if there is a absolute problem with this.

This is the site of a Luxemburg bank that does mention non-resident
account-holders and will be aware of any problems for US applicants.
Of course, there is a contact facility on which you can ask:

http://www.bgl.lu/SitePublic/ServiceServlet?PAGE=HomePage&

If you search with   online bank  you will find many sites.

This is one from an English bank.  If you scroll down you can click on
"Personal banking terms and conditions (PDF, 175kb)"
and get a complete outline of account conditions:

http://www.hsbc.co.uk/1/2/personal/current-accounts/bank-account

I hope this is of assistance to you.  If you have questions, please let me know.
It is getting late now in Germany and I cannot make a couple of phone
calls that might add information about the possibility of this
solution.

Myoarin