Q. A continuous function on [0,1] is chosen at random, what is the probability that it is differentiable?

A. Do not ask it this way. You have to specify a probability distribution on the set of these functions.
A finite set carries a distinguished probability measure (all points of equal probability), invariant under bijections (permutations), but an infinite set does not. A point of a continuum cannot be thought of as a unit of measurement! See also Bertrand paradox.

Q. A random variable is distributed uniformly on [0,1]; can it take the values 0 and 1? I understand that this event is of zero probability. So what? Can it happen, or not?

A. Do not ask it this way. Probability theory calculates probabilities; that is all.

Q. A random variable is distributed uniformly on [0,1]. What is the conditional probability that it is equal to (say) 1/3, given that it is a rational number?

A. Do not ask it this way. Conditioning on a negligible event (that is, an event of zero probability) is a subtle matter; sometimes it is possible (in some sense), sometimes not. We'll consider it in detail in the course.