Answer: Hugh Judson Kilpatrick, a member of the class of May 1861, was wounded in the Battle of Big Bethel on June 10 of that year. He thus became not only the first member of his class but also the first U. S. Army officer to be wounded in a Civil War battle.
Sources:
https://en.wikipedia.org/w/index.php?title=Hugh_Judson_Kilpatrick&oldid=842688737
Wittenberg,
We Ride a Whirlwind, p. 54
bonus
Answer: The
First US Cavalry, organized by a March 3, 1855 act of Congress and later re-designated as the Fourth US Cavalry. "
The military fitness of those selected for the First (now Fourth) Cavalry is indicated by the high commands to which many of them rose, as follows" https://history.army.mil/books/R&H/R&H-4CV.htm
Edit - For the main question, a lot of people answered Adelbert Ames, perhaps under the impression that Judson Kilpatrick was not a member of the class of May 1861. The following source, cited by at least two players, shows that he was a member of that class, though for some reason his name is preceded by Kingsbury and Kirby, who should come behind Kilpatrick if the list were strictly organized in alphabetical order.
http://civilwarintheeast.com/west-point-officers-in-the-civil-war/class-of-may-1861/
For the bonus question, I am accepting Fourth Cavalry as well as First Cavalry, since the unit was known by both names at various times. I am also accepting Second/Fifth Cavalry, which has been supported by a source provided by HarlechMan.
And, since the question seemed to be emphasizing that the correct answer would be a regiment that provided a large number of future Civil War generals, I will also accept the First Regiment of Artillery, which actually had 40 future Civil War generals among its ranks at one time or another, way more than either the First or Second Cavalry.
hoosier