Suppose you start with a "randomly-chosen" NFA M of n states. (There are many possible definitions, but let's say we choose some real number p and put an edge between any two states with probability p.) How many states do you expect to get, on average, when you do the subset construction on M? Nobody knows.
Other kinds of DFA's we did not study in 360 include the 2-way DFA and the 2-way NFA. Sometimes we cover them in 462.
I mentioned an open problem in this lecture: consider the language L of primes expressed in base 2. Is L* regular? Nobody currently knows, although everybody believes the answer is "no". If you are able to solve this, you get an automatic 100 for the course.