In Answer to the Eureka Phenomenon by Issac Asminov

Question 1
Does Asimov argue that science ought to abandon reasoned thought in favor of intuition?

No, I do not believe his argument is that there is no need for reasoned thought. I think instead this is what he believes:
Imagine the rigorous thought process as a group of slave workers who are digging in search of the tomb of a long lost king. They totally uncover the tomb, but they see mysteries that they cannot decipher, and then the archaeologist comes in, and he opens the tomb.
He does not refute the importance of thought, but he does believe that intuition, as is used in the question, is responsible for more discoveries than the doctrine of science presumes. (See page 207, paragraph 4)

2
What does Asimov find wrong about scientific reports as they are customarily written? Do you agree? If scientific writing were not strictly reasonable, wouldn't there be a danger of misrepresenting science?

He is disappointed that very few scientific reports include that “Eureka phenomenon” he believes is responsible for a majority of the accomplishments and discoveries that scientists make. Poincare makes a similar, albeit converse, argument which is quite sensible; it must since his thoughts on the matter are acclaimed as one of the most important articles ever written on creativity. Poincare explains the circumstances behind the completion of his Fuscher function theorem: where “ideas collided until the pairs interlocked” involuntarily after he had mentally exerted himself seemingly to no avail. Then, he had to work out that inspired thought, which he agreed had become considerably less difficult, before he had any considerable success.
I believe that contemporary science reports are boring and lack imagination. I also believe that very few extraordinary discoveries can be made by extreme mental exertion, because I do not believe that man even understands the mind enough to control it. I agree that the amount of work put into a research should be penned down and the researcher deserves credit for his industriousness; but I believe it is ingenuity and genius for a thought to unconsciously reveal itself to a person. I believe that science is afraid of the unknown, and the relation this may have to the supernatural.
To the last section of the question, I ask “what is science?” In my opinion, science should be the act of finding out, irrespective of the repercussions that this may have on the dogmatic beliefs of the intellectual caucus. I don’t believe that science should be afraid of being misinterpreted because it is an indefinite, widely encompassing and largely unknown area which must be open to foray by any and all means possible, as long as those means are morally proper. Again, the definition of moral can become largely ambiguous, so I define it in terms of the two greatest laws on earth: “Thou shall love the Lord thy God with all thy heart, soul and mind” and “Thou shall love thy neighbor as thyself.”

3
Is cultivation of the "Eureka phenomenon" encouraged in any of he science courses you may have taken or are now taking? Why, or why not?

I honestly do not believe there is a definite answer to that question, because the school system seems to be designed to teach students how to teach themselves, through a process of the school teaching the students. This is almost a rigmarole method, which is sometimes unnecessary, judging by the success of college drop-outs like Bill Gates, the Steves: Jobs and Wozniak, Ted Turner and many others. Also, the teaching process is dependent on the professor, and I do not believe any of my Science classes this year necessarily advocate the cultivation of the “Eureka phenomenon.” Then again, I believe it is the discretion of the student to apply his knowledge outside of the classroom in constructive and unique ways, (maybe by joining a research) , in order for the mentioned phenomenon to be put to the test.

4
Have you ever experienced anything like the "Eureka phenomenon" Asimov describes? If so, write out an account of what happened. Tell us what your feelings were when the phenomenon occurred. Did you ever report the discovery in just that way to any one else?

The “Eureka phenomenon” has definitely occurred countless times in my life. It can come from the influence of certain types of music, or it may be from a particular scene from a movie. Or it may just be from reading my Bible and talking with God. It is my belief that all these things are still tied, in some inexplicable way, to my conversations with God, and they are merely an offshoot of those divine moments.
I have not necessarily reported them to anyone in particular. I engage in ordinary conversations with random people, conversations that mostly turn deep, when I confess my testimony of God and my faith.