上司に残業しても良いか尋ねよう

1,2. Your expression was correct, but since this is a conversation role play, you'd better write "from A to B" for more clarity.
3. We use "today" to refer to "this day."
4. You cannot just say "if you permit" because "permit," in this meaning, is a transitive verb, which means it takes an object. You can, for example, say "If you permit me to do so" or "if you give me permission."

1,4. Use the present perfect tense to refer to a past action with an ongoing result. You had that experience in the past, and that experience is still with you, so you need to say "I have had the experience of doing that before." However, you can simply use the present perfect tense to talk about things that you have experienced in life, which means you can just say "I have done that before."
2. You cannot use the definite article "the" because you are not talking about some specific permission you have already mentioned or known to your listener.
3. We always get permission FROM someone just as you, for example, get a gift from someone.

1. The proper conjunction here is "which" because the following clause "is named CrowdLog," is a nonrestrictive clause, which means it is not an essential part of the sentence, but just some additional information. We use "that" for restrictive clauses.
2. Aside from some fixed expressions such as "I understand" or "Do you understand," we often use "understand" as a transitive verb, which means an object is usually necessary. That's why I added, "the system" as the object of the sentence.