Your statement that "Jude is not referring to the Book of Enoch" has no basis in fact as far as I can tell. It is merely an assertion, and it contradicts all scholarship on the topic. Read any competent scholar and you will find them confirming this fact.
Now moving on to your questions:
What specific incident is Jude reminding his readers of?
He is referring to three incidents:
1) The entire 40 years in the wilderness, when God destroyed all those who did not believe him. "And the LORD'S anger was kindled against Israel, and he made them wander in the wilderness forty years, until all the generation, that had done evil in the sight of the LORD, was consumed" (Num 32:13).
2) Genesis 6:4 "There were giants in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown."
3) Sodom and Gomorrah
Who or what groups of people were destroyed in the Wilderness after deliverance from Egypt?
All who did not believe.
What are the names of the ministers destroyed?
Jude made no mention of any "ministers" in the verse you quoted. Your interpretation is forced and it contradicts context. Jude refers to Michael the Archangel in the immediate context. He would have been entirely incompetent to refer to humans as "angels" just two verses earlier. If you want to make your case, you must first establish that it is reasonable to think that the "angels" in verse 5 really refer to human "ministers." Why haven't you done that?
By what means were the ministers destroyed?
Again, the text you quoted made no mention of any ministers. It specifically spoke of God destroying all those who did not believe in the exodus.
All the best,