Who (if anyone) owns the rights to Mark Twain's/Samuel Clemens'
image and likeness? Am I allowed to use him as a character (e.g., in a 
comic book, movie, or video game) in a piece of fiction that has little or no
historical veracity aside from his name and some notable quotations
without seeking approval from some estate? If an estate does exist and has
control over his likeness, what would constitute fair use or parody in this case?

If there is some control over the use of his image/likeness, I want to
know who I have to contact. Either way, I would like precedent cases
(such as his "appearance" on Star Trek: The Next Generation, and his
role in one of Neil Gaiman's "Sandman" comics), though I should stress
here that in this particular case the goal is to cast him as the
central character and not as a secondary/tertiary character.

Hello and thank you for your question.

Yes, you are allowed to use him as a character (e.g., in a 
comic book, movie, or video game) in a piece of fiction regardless of
historical veracity without seeking approval.

For example,
"Public Domain Images is a commercial enterprise that offers some
interesting and freely available jpeg files. They also provide a
thoughtful and concise explanation of copyright considerations on the
Web. The photographs offered for free downloading include a variety of
scholarly topics, drawing from the sciences and the humanities. If you
need a picture of DNA or a portrait of Mark Twain, look here first.
They ask that you include a credit link back to their site. Access:
http://www.PDImages.com/
http://search.cartserver.com/search/search.cgi?maxhits=1000&cartid=a-6896&category=a&keywords=Mark+Twain&go=GO%21

"These images are believed to be copyright free because their
copyright has expired [which will be true of any pre-1923 image], the
Copyright owner has lost or given up their Copyright, or they are US
Government material free of copyright by law."
http://pdimages.com/c/h-cadvise.htm

How to Know if Copyright has Expired
"According to the latest revision of the US Copyright Act, all items
copyrighted in the US before 1923 are now public domain. Items
copyrighted between 1923 - 1963 have expired if they did not file a
copyright renewal. The way to find out if a 1923 - 1963 work renewed
is to do a search (or hire someone to do a search) at The US Copyright
Office in Washington, DC. Unfortunately, the records for copyright
registrations and almost all renewals for these years are not online,
and must be looked up in paper records. On most of our images we have
already done this work for you, and the results are noted in the
Restrictions box for each image."
http://www.pdimages.com/c/h-cexpired.htm

Other Laws Concerning Images
"In total, there are four layers of ownership in every image, and
copyright ownership of the image is only the First Layer. Public
domain, or copyright-free images can still be subject to laws of three
other layers of ownership.
Layer Two concerns ownership of the physical copy of the image, and
any restrictions imposed by the owner or source. For example, we
require a web site link or photo credit to "PDimages.com" when any of
our images are used.
Layer Three concerns ownership of the subjects portrayed in the image,
including copyrighted subjects, trademarked subjects, and other
protected people, places, things and events.
Layer Four concerns how you will use the image, whether commercially,
editorially, or in advertising. Your use may also touch on such legal
areas as right to publicity, right to privacy, misrepresentation,
defamation, libel, or tarnishment.
Only by satisfying all four layers can you use an image legally.
Obviously it is not possible in this short help screen to cover all of
the areas of concern, but in general, users of images from the
PDimages web catalog do not have to worry about Layers One and Two
(read the restriction notes for the images you are using to be sure).
Layer Three is mainly of concern if you are using an image of a famous
person from the last hundred years or so, and if you are using them in
a commercial or advertising way (Layer Four).
For example, putting a copyright-free image of Marilyn Monroe on a
T-shirt would be a commercial use, or using her in a magazine ad would
be an advertising use. Both commercial and advertising uses would
require payment to her estate even though the image itself is public
domain. Using that same image inside a commercially published book as
an illustration would be an editorial use, and would not require
payment."
http://www.pdimages.com/c/h-other.htm

So impersonations or images of Mark Twain in Star Trek, Sandman, etc.
as literary uses were perfectly legal without permission.

For precedent specific to Mark Twain, consider this review of
Lighting Out for the Territory
Reflections on Mark Twain and American Culture 
http://www.oup.com/us/catalog/general/subject/HistoryAmerican/Cultural/~~/c2Y9YWxsJnNzPWF1dGhvci5hc2Mmc2Q9YXNjJnBmPTkwJnZpZXc9dXNhJnByPTEwJmJvb2tDb3ZlcnM9eWVzJmNpPTAxOTUxMjEyMjg=
"'An energetic report on how Twain's attitude toward race developed,
how his works have been used and abused, and how the image of himself
that he so carefully invented has been coerced into making guest
appearances in other people's fiction, movies, plays, even 'Star Trek:
The Next Generation.'"
Id.

Search terms used:
"public domain" image 
"star trek" "mark twain" clearance
"public domain" image "mark twain

Thanks for letting us help.

Sincerely,
Google Answers Researcher
Richard-ga

Thanks. I appreciate the thouroughness.

As a historical figure, no one "owns" Mark Twain. Not his writing, nor his image.

Kriswrite

Well I know that at least some of his post-humous work is still under
copy protection, with rights reserved and the like... can you provide
documentation to the effect that his image is not protected?

thanks...

Because of the era in which he wrote, his work is in the public
domain. A publisher may only copyright the format in which they
present his work. (For example, you couldn't make an exact copy of a
modern book layout, because the layout is protected.) For more about
public domain, see: http://www.unc.edu/~unclng/public-d.htm

Kriswrite