Question 1: The example Have the Olympic games began? is incorrect. The perfect tense of to begin is begun, for all numbers. Have the Olympic games begun? is correct.
Question 2: Both are grammatically correct. Began is the simple past tense of to begin for all numbers, and begun is the perfect tense of the same verb, as in Question 1. These two usages do express exactly the same thought from the same perspective in time, but beware of generalizing this case. There are expressions in which the past and the perfect do not express the same thought. A good example is:
What did you do at school today?
This would be perceived as merely a request for a synopsis of the day's events at school. Contrast it with:
What have you done at school today?
This would be perceived as attaching importance, and probably some notoreity, to the listener's actions at school.
The perfect tense deserves your close attention as a student of English. A frequent Answerer at ELL has posted a very useful resource here: What is the perfect, and how should I use it?