The domain (if not given explicitly) is all real numbers x,y for which the formula makes sense. There are two things that usually limit the domain: you can't divide by zero and you can't take the square root of a negative number. So for (c), .
The range is the set of all values the function can take. For (a) and (b), you can find x and y so that z is any real number. For (c), since it's the negative square root, (the negative square root of zero is zero).
For sketching the graphs, hopefully you have some experience working with various functions and know what some of these look like. If not, you can just plot a bunch of points.
I think by traces you mean the curves where z is constant. So if you think of z as a constant, your experience should tell you that (a) is a line, (b) is a hyperbola, and (c) is an ellipse.
Hopefully that gets you started.