Suppose we generate two uniform random numbers X and Y from [0, 1] independently. (1) What is the sample space?

A sample space is the set of all possible results of a random experiment.

Let the random experiment be to choose a number in the interval [0, 1] and let X be the value chosen in that experiment.

On the other hand, whether the random experiment is to choose a number in the interval [0, 1] and let Y be the value chosen in that experiment.

Since the two experiments are independent, the sample space for the joint random variable (X, Y) is given by the cartesian product [0, 1] X [0, 1]