Countability of disjoint intervals

Your thinking is correct; the set of all singleton sets of R is certainly uncountable.

It seems that the question meant something like "Every collection of disjoint open intervals in R is countable." (In this case, the claim that each interval contains a rational number is valid.)

Maybe there was some convention in the course that "interval" meant open interval, or excluded singleton sets; perhaps it's simply a mistake. Either way, it's good that you noticed this detail!

Because "singleton interval" is usually not considered to be an interval.