Was he familiar with Twain?
While I really couldn't find much on this topic, here is what I found. As a comment mentioned,
there's Clyde Kilby's observation that he "was pleasantly surprised at the familiarity he showed with American literature, especially that of Mark Twain" (TOLKIEN & THE SILMARILLION, pages 30-31)
from this website; the same however also points out that what was believed to be a reference to Steinbeck was in fact not, and a comment on the blogpost by Hammond and Scull, two of the foremost researchers of Tolkien and authors of (among other works) the massively definitive J.R.R. Tolkien Companion and Guide, said:
Wayne has read Mark Twain, whose books were required in American schools, while Christina's knowledge of them from an English upbringing is second-hand; and talking about this, we agreed that although neither of us has read a word of, say, G.A. Henty, we both recognize titles such as With Clive in India as typical of his works, and we've read enough about British children’s literature to know what kind of books Henty produced. Which in turn made us wonder how much Twain Tolkien had actually read when he spoke with Kilby, as opposed to having enough familiarity through other avenues to carry on a casual conversation.
Which also seems to cast doubt on Tolkien having known of Mark Twain. Then, I found this little gem, a transcript of a letter by J.R.R. Tolkien to Naomi Mitchel which seems to clearly indicate that Tolkien was indeed aware of at least some of the works of Mark Twain - the letter contrasts Michel's book, The Chapel Perilous, and A Connecticut Yankee in King Arthur's Court:
I hope that the Chapel Perilous went and is going well [though I seldom see or read reviews, and but for a cutting agency should not see my own – In spite of them I do not read them all] A curious and rather disturbing blend. I hope you won’t mind my mentioning “a Yankee at the Court of King Arthur.” I only do so, because of the shattering difference. Of course there are the deliberate
mechanical anachronisms in both. the hello–girls and telephones, and electrified wire; and dwarfs with photographic apparatus: I don’t like ‘em, of course, and think it better if the satire arises out of the material as it is; but there are umbrellas in the Shire—but M.T. [Mark Twain]was merely vulgarly romantic, whereas you undermine, or (perhaps fairer) transform the whole thing.
See this site.
With Joan of Arc?
It should be noted that Joan of Arc was not one of Twain's more...popular works. To quote Wikipedia on the matter,
Twain's opinion notwithstanding, critics, then and now, have not labeled Recollections his best work. Today, the book is hardly read or acknowledged in the mainstream, especially compared to Twain's comedic works, such as Huckleberry Finn, Pudd 'nHead Wilson, and Tom Sawyer.
(A Connecticut Yankee in King Arthur's Court could probably be added to the list of "more popular comedic" works.)
On the other hand, at the minimum, Tolkien was probably familiar with the legends of Joan of Arc. This paper for example, compares the character of Eowyn to Joan of Arc and cites multiple others that do the same. In the digging I've done up to this point, a bunch of sites cite (sorry, bad pun) Eowyn as a Joan of Arc figure, and as a reader of the stories, I do have to agree that that seems like a pretty clear analogy (though on the other hand, if you're trying to write a female character that "does something" into a medieval society, a Joan of Arc character is kind of what you're going to get at first pass; take that how you will).
What is also of note is that Twain based the book off of actual sources of the story of Joan of Arc - to quote the book itself:
Authorities examined in verification of the truthfulness of this narrative:
J. E. J. QUICHERAT, Condamnation et Rehabilitation de Jeanne d'Arc.
J. FABRE, Proces de Condamnation de Jeanne d'Arc.
H. A. WALLON, Jeanne d'Arc.
M. SEPET, Jeanne d'Arc.
J. MICHELET, Jeanne d'Arc.
BERRIAT DE SAINT-PRIX, La Famille de Jeanne d'Arc.
La Comtesse A. DE CHABANNES, La Vierge Lorraine.
Monseigneur RICARD, Jeanne d'Arc la Venerable.
Lord RONALD GOWER, F.S.A., Joan of Arc. JOHN O'HAGAN, Joan of Arc.
JANET TUCKEY, Joan of Arc the Maid.
So it could very well be that Tolkien was not in fact familiar with this particular tale of Twain but was familiar with the source material from which Twain drew. On the other hand, these sources are primarily in French, and to quote letter 213:
For instance I dislike French
So I'm not sure he would have necessarily read these sources which are mostly in French, it seems.
Yes, he was probably familiar with Twain, and he might have had a passing familiarity with the story of Joan of Arc, but he probably did not have the level of awareness necessary to have Theoden be influenced by the Dauphin/Joan of Arc, and even if he was, Twain was probably not the transmitter in this case.
Of course, all of this must be taken with a grain of salt; see Rand al'Thor's comment.