I want more information on the Karaites, and their relationship with
the time around Mohammed and his interaction with the Jews in Medina. 
I am interested in historical and sacred histories.

Hello there

I suppose I should begin by saying that the chance of Mohammed's
interaction with Karaites in Medina is slim to none.  The reason being
that Mohammed was in Medina in the seventh century and the Karaites
were founded by Anan ben David in the eighth.  Rather than the
Karaites having an influence on early Islam, it is the other way
around.

"The founder of Karaism was Anan ben David, a prominent Jewish scholar
in eighth century Babylonia. His opposition to rabbinic authority was
due partly to personal rivalries within the establishment, and partly
to his recognition that Talmudical Judaism was going through a period
of stagnation at the time and might be revived by the stimulus of
renewed Biblical and Hebrew research. Influenced by Islamic
scholarship, his achievement, paradoxically, was to inaugurate a new
epoch in the history of orthodox rabbinic Judaism which was to
culminate in the work of the great mediaeval Jewish scholars. The
Ananites, as his anti-rabbinic followers were called at first,
included some of the greatest Jewish personalities of the day. They
were renamed Karaites in the next century by Benjamin of Nahavend, and
could have become a major force within Judaism had it not been for the
counter-offensive led by Saadiya, the greatest of the Geonim
(882-942). Thereafter Egypt was the chief centre of oriental Karaism
until it was weakened by the authority and reputation of Maimonides in
the twelfth century." - Quote from the division of Religion and
Philosophy, St. Martin's College -
http://philtar.ucsm.ac.uk/encyclopedia/judaism/kara.html
Their homepage is  - http://philtar.ucsm.ac.uk - This is a very large
website where you can find information about most any faith and the
page I took the quote from has a full outline of Karaites history.

As for who they were (and are) - The Karaites are still with us. 
There are about 1500 in the USA, about 100 families in Istanbul, and
about 12,000 in Israel, most of whom live in or around Ramleh.
(information from the same source as above)

The Jews in Medina probably did much to prepare the way for the
teachings of Mohammed.  When Mohammed first went to Medina, he was
friendly toward the Jews and they were included in a treaty between
him and the inhabitants of the city.  He also made concessions to them
on the ground of religion.

However, the Medina Jews took great delight in taunting him,
ridiculing him whenever they could and attempted to draw him into
arguments to try and expose his ignorance.  His friendly attitude was
soon turned to enmity.  A few Jews did convert to Islam, including
Abdallah ibn Salam, whom Mohammed called the "Servant of God."

Finally, Mohammed resorted to violence against the Medina Jews.  This
chapter of history has so many stories about who insulted whom, who
attacked whom, what happened in different parts of the city etc,  it
would only create confusion to list them all here.

The end result is that the Jews of Medina were exiled or destroyed.

The above information taken from Jewishencyclopedia.com.  The page
pertaining to Medina will give you the details of the various stories
I mentioned above. -
http://www.jewishencyclopedia.com/view_friendly.jsp?artid=326&letter=M

When it comes to both the profane and sacred histories of the
Karaites, we once again have innumerable resources just within the
Jewish Encyclopedia itself.  So the best thing I can do for you is to
point you to those resources and allow you to make your own judgements
as to which you think appropriate.
http://www.jewishencyclopedia.com/search_results.jsp?searchType=1&pageNum=1&search=Karaites&searchOpt=0&x=12&y=9

There are 50 different listings dealing with history, personalities,
teachings and scriptures.  Since much of this material is subjective
in nature, I doubt if you would want a researcher trying to digest it.
 Such a process is liable to put a spin on things depending on our own
philosophies and background.  We are a widely varied bunch and would
approach such things with an equal variety of interpretations and
opinions.

Between the two resources I have named, you should be able to find
that the bulk of what is known about the Karaites will be covered.

Search - Google
Terms - karaites, karaites history, jews of medina, medina jews
mohammed

If I may clarify anything, please ask.

Cheers
digsalot

Very complete answer and was scholarly and incisive.