Here's a fairly comprehensive (though by no means complete) list of banjo tunings:

http://zeppmusic.com/banjo/aktuning.htm

It includes such tunings as dADAD, aADAD, aDAAD, aDADF#, and others. (in written banjo tunings, the first lower case letter is the short drone string)

The real question however is what you want to do by tuning a banjo to D and A notes.  What we think of as "dulcimer tunings" are really just notes. Usually a tuning is used to achieve a specific purpose.  It enables us to play a tune more easily, or to play it in a certain way or with certain fingering positions.

You haven't told us what your situation or your goals are, so I'm making assumptions here-  As a longtime player of both mtn dulcimers and banjos (and I'm really more a banjo player than dulcimer player), my personal advice would be this-  play your dulcimer like a dulcimer. If you want your dulcimer to have a tone like a banjo, get a dulcimer with a banjo head on it.  Learn to play and (initially) tune your banjo as a banjo, rather than trying to make it imitate a dulcimer.  Banjos are designed to be played most easily like banjos, and learning to play a 5string oldtime banjo is not all that hard, after getting past the first little learning hump of unfamiliar right hand motions. IMHO if you're tuning a banjo like a dulcimer in order to try to make it easier to learn to play (if you're a dulcimer player), you are doing yourself no favors.  It might make the first week or two seem easier, but it will make playing the banjo harder and way more limited later on, plus most banjo learning material will then be useless to you as well. 
If I've made incorrect assumptions, then forget all this.  lolol! 

Probably. I would imagine almost every tuning has been tried by somebody. An instrument with 4/5 tuners [or more] could be tuned [with string research] to just about any tuning using some, or all, of the tuners available. Would it be satisfactory? That would be up to the player.  I play a 5 string Blue Lion with only 3 strings installed and it is great.

The name applied to the tuning probably depends on which instrument is being discussed. 

I've used a "D" tuning on my banjo and played the melody on one of the D strings using the rest of the strings as drones. It is not ideal, but works.

Ken

"The dulcimer sings a sweet song."

Banjos have five strings, including a short drone string.  Oldtime banjos can be tuned to over a hundred different tunings, in many many combinations.  Some of those combinations can include strings tuned to D and A and yes also DAd and include other notes on other strings, like gDAda. Keep in mind that our familiar DAd or DAA dulcimer tunings are nothing more than a tonic note with a fifth... and "tonic + 5th" notes appear in the tunings of all kinds of other instruments as well, since it's a common musical interval to use.

Banjos have lots and lots of tunings.  The reason people tune banjos in different tunings is usually in order to play certain tunes or certain 'families of tunes' more easily.  Since it can be hard to remember so many tunings by naming all the tuning's notes, some banjo tunings are conveniently referred to by a song name associated with it... such as "Sandy River Belle tuning", "Little Birdie tuning", or "Reuben's Train tuning".

Paula -- it can be done, as Robin suggests.  Probably better with a banjolele or other short-necked version than a full size instrument.

Paula, I have a Dulcijo-- a little 3-string banjo tuned to DAd.  To play out of G, I'll re-tune the middle string to G.  I haven't had much chance to play around with it yet hope to in the coming year.  Here is a photo: 

Has anyone ever retuned a banjo into  mountain dulcimer tuning?