What is the average waiting time of HTHH and THTH?

Answer: 18 and 20

What is the probability that THTH beats HTHH?

Answer: 9/14

Is it true that a sequence with a shorter expected waiting time must be more likely to occur before a sequence with a longer expected waiting time?

Answer: No, you have just seen the counter-example.

One of the key tasks in your degree is to help formulate these questions and ideas precisely.