Yes, velocity is the time-derivative of position, and acceleration is the time-derivative of velocity. X is definitely horizontal position and Y could maybe be vertical position, but that’s often ignored in favor of just using theta to describe the angle. Neither one is a velocity without a dot.
Now that I’m thinking of it, the phase portrait is likely to be a visual relation between the varied pendulums, with Y being a function of the length of the pendulums. Think of all those ratios coming together at 30 seconds – that could be when they cross the y-axis! The word phase usually means the difference between two angles.
Sometimes it’s useful to see how something changes not with time but with distance. For instance, dy/dx is usually how height changes with length. In order to find the smallest length of thread needed to move from point A to point B, you derive by distance.
There are other uses, too. Usually a derivative is used to find the minimum or the maximum one can use of something.