Question: I've started to teach myself how to play the piano, and one of the difficulties I've found is trying to get both hands to do things at once. I either forget one hand entirely, or have one hand trying to mimic the other. 

This reminded me of many years ago when I was taught to play the viola. For many weeks, we did not use the bow, instead focusing on getting the left hand to do the things it was supposed to, and learning to read music. In hindsight, this makes sense to me, as it's often important to get used to some fundamentals before attempting to do everything at once. 

So I was wondering, when learning the piano, is it better to learn each hand on their own, or muddle through the difficulties of synchronizing them from the beginning? Or, if there are arguments for both methods, what are the pros/cons of each? What problems may develop, or what difficulties might be avoided? Or is there perhaps a happy medium that has proven the most successful?

For my personal musical background, I have about nine years' experience with the viola, though I haven't played it in about five years (stopped after high school). I also play the guitar, but never learned to read music for it.

Answer: 
When I first started learning piano I also used to practice one hand at a time. I think it is important because then in the end it's all about coordination and if you want your hands to be well coordinated you first need each hand to know what it has to do.
Putting it all together is then a completely different story, you'll have to focus on the coordination and not on the single hand anymore plus you'll have to read 2 staff at a time (which might be tricky for someone which is not used to that), but indeed the practice that you gained playing with separate hands will help.
With experience and practice you'll do that more quickly and start putting it all together soon, however I still give a shot at each hand separately, I think it helps a lot. Of course with increasing difficulty of the piece you want to play you'll spend much more time practicing with two hands together. At times you'll have to focus on a small amount of bars when you'll have some difficult phrase (and then there are many ways which might help studying a single piece depending on the technique involved), which is also recommended when there are some particularly difficult parts in the piece (but I think this applies to the viola as well).
I have never heard of one-handed practice actually being *harmful* for any piece of music, nor that it might teach you something you would later have to unlearn.

Slow practice and one-handed practice have both been essential to me on the piano, (although separating the limbs doesn't seem to help me on drums, interestingly), and one thing that helps very much in combination with them is dividing a bar up into a larger number of sub-beats, to understand it.

Imagine trying to tap out 2 beats per bar evenly on one hand, and 3 beats per bar evenly on your other hand.

If you subdivide the bar into 6 then one hand takes beats 1 and 4 while the other takes beats 1, 3 and 5. Many rhythms can be understood more easily that way, and practiced more slowly, on pretty much any instrument.

Large books full of short, graded exercises will not only strengthen your little fingers but prepare you for all sorts of confusingly counterpointed melodies.