Sudoku: a rather more interesting question

I watched someone on the Metro this morning doing a Sudoku puzzle. Personally I find them dull in the extreme, an entirely algorithmic and mechanical process that hardly exercises the brain at all. They are nowhere near as much fun as a good cryptic crossword. They do, however, present a couple of interesting features that I have pondered and would genuinely like to know the answers to.
The first, and probably least interesting, problem is how are they created? Presumably a full grid is created and then some numbers removed but how is that grid created? How do you start with an empty grid and ensure that as you write the numbers in you don't write yourself into a position where it is impossible to fill the remaining squares. I assume there is some kind of algorithm that allows this to be done but I'd like to know what it is.
The second puzzle I've pondered is this, how many different grids are possible? I have tried to devise methods of working this out, and usually I'm quite good at probabilities and combinations but I haven't been able to even think of a way to approach this problem.
The third, final and most difficult is this - starting from a full grid how do you remove numbers to arrive at the minimum sufficient condition for a single solution? Remove too many and there will be multiple solutions, don't remove enough and it will be easier than you want. How do you make sure that what is presented contains enough, and only enough, information?
The Sudoku puzzle itself I find remarkably boring but I really would like the answers to these questions.
Anyone got any ideas, or a source that shows the answers?