I have a theory about why someone often gets on a trike, and then instantly turns the opposite way they mean to when they first try a turn, something I've seen a lot (and done myself, at first):
On a bike, you lean into the bend. On a trike, you can't. So your brain is expecting you to be at an angle when you turn, and you're not. So your brain thinks you're leaning the wrong way, and instinct makes you overcompensate, hence suddenly turning the other way.
And yes, it's funny how many times you have to tell some people not to put their feet down when they stop.
I think our work Maximuses are two wheel drive with a diff. I've never yet had the nerve to get one up on two wheels - I've done it with a Pashley Picador type though. That was one wheel drive, so to sustain any distance on two wheels, you have to get it up on the right one, or you pedalled and pedalled and just slowed down.
I watched some grass track trike racing at the CTC rally some years back, and it looked way more exciting than the bikes - perhaps just because of the amount the riders were leaning in.
As to why - well, I ride a recumbent trike instead of a recumbent bike, and I like it so much because of the stability. I ride upright bikes though, easier to store and get out day to day.