Rob basically answered your first question, but I'll just add a little bit more.
Optical zoom actually changes how the light hits the sensors on the camera, and therefore you get different light hitting different parts of the sensors when you zoom in/out.
Digital zoom, on the other hand, doesn't change what light hits the sensors, it just changes how the information is saved. So the processor basically just does a digital enlargement of a portion of the sensor data (with some interpolation to fill in the gaps).
The reason you wouldn't want to use digital zoom is because it lowers the quality of recorded image (because the interpolation isn't really matching what you would see in real life). You can always get the exact same results as a digital zoom (if not better), but just resizing/cropping the image after it is recorded. Even the most basic image editing program can do that. That way you can still have the original file in its best possible quality, and still achieve the 'zoom' in post if you really want that.
Digital zoom is more practical in digital video cameras (although it still lowers the quality of the image), because it is much harder (at least for hobbyists) to zoom the image in post production for a video than for a still photo.
As for your second question (I know this post is already getting long, but I'll try to be brief). What post processing you do, and how much you do, all depends on what you are trying to achieve. If you are trying to just do photography that looks its best, or if you are trying to create digital art.
If you just want to touch up a photograph, then you would probably stick with the basics on most of your shots: colour, contrast, and level adjustments, and possibly sharpening/softening. You might also use noise reduction depending on how grainy your shot is.
Of course, there is a lot more you can do if you really want to spend the time and get everything exactly the way you want it. And if you want to get into digital art, then you can basically manipulate every aspect of the final image, by combining multiple images, using filters, etc.
But I would generally say that most good photographs have some adjustments made to them, even if you are basically just using the computer to reproduce what you would adjust in a dark room.