I need the names/contact info. for several (at least three) of the
best website designer firms in the US.  If they have a lot of
experience with real-estate-oriented sites, that would be a plus. 
Ideally, the websites of these firms will contain samples of their
work.

What you should really want is a good honest and talented individual
working from his house. These people exhist, but the can be tricky to
find. Try typing "website design" into a search engine, but start your
browsing on the 5th page of results. The "firms" who occupy the
coveted positions on pages 1-4 are way too busy to do an honest hour's
work for you.

patrice29-ga is 100% right.
I myself work as a dedicated esigner ;) for a US based company, i do
all the online work needed (avg. 3-6 Hrs./day) and they pay me well.
Well for them because they pay me much less that US prices to really
have your own designer. And well for me because that converts to a
very good amount of cash in where i live ;)

bookshelf,

Explore all the Real Estate websites you can, look at hundreds, find
one you like, that is a fast site, well designed, with all the bells
and whistles you want, and scroll to the bottom... 9 out of 10 times,
the web site designer will have his contact info at the bottom of the
home page, or the bottom of the contact page.

Rather than choose a web designer or agency that will deliver,
everything you as a client would like to see how about you choose
someone with experience in the customer journey and perception?

This may sound like "Hokum" but designing your site with a user in
mind and around the most common task they would be looking to
undertake or fulfill would ensure that your rate of return is higher.

Many companies are out in the industry looking to make a sale but if a
customer believes from the very beginning that they are valued and the
information contained within a site relates to them they will return
or continue.

Before you spend any money on design, consider what you want your site
to do and how quickly it can be done.  Once you have that information
approach several designers to see is they really understand what you
need.