Is reflection or rotation in a $2$-dimensional normed space isometric? Is reflection in the $x$-axis or in the line $y=x$ in a $2$-dimensional normed space isometric? How about rotation through a right angle? If so, what is the proof?
 A: You don't say what norm/metric you're using, or whether it is allowed to vary. So, I'll assume for now that you're using the conventional Euclidean metric: $d\left((x_1, y_1), (x_2,y_2)\right) = \sqrt{ (x_1 - x_2)^2 + (y_1 - y_2)^2 }$.
Then, since we're dealing with the space that we all live in, we can trust our intuition, and the answers should be obvious, geometrically -- reflecting things or rotating them is not going to change distances. Maybe this doesn't qualify as a "proof" (though, personally, it's good enough for me).
If you want a formal algebraic proof, proceed as follows:  take two arbitrary points $P_1 = (x_1,y_1)$, and $P_2 = (x_2,y_2)$. Their reflections in the $x$-axis are $Q_1 = (x_1,-y_1)$, and $Q_2 = (x_2,-y_2)$. You have to show that the distance $Q_1Q_2$ is the same as the distance $P_1P_2$. That shouldn't be too hard.
The other two examples can be handled with the same sort of approach -- take two points, calculate their images under the mapping, and then show that the distance between is unchanged.
However, if we're allowed to use any norm we like, then things get more interesting. For each of the three given mappings, you can find some norm on $\mathbb R^2$ for which it's not an isometry. But, my guess is that this is not the question you're asking.
