The screen appears scratched when it is examined carefully.
Will it be also grammatically correct if I removed “it”?
The screen appears scratched when examined carefully.
Yes, but I think it sounds better reversed:
When examined carefully…
No, it would be grammatically correct if you remove “it is”.
The screen appears scratched when examined carefully. I am not sure though.
You removed “it is,” but the result is correct.
Not how it is.
You could put… When examined carefully, the screen appears scratched.
I can’t place “when examined carefully” at the end??
The screen appears scratched when examined carefully.
Please, is this sentence (exactly as it is below) grammatically correct.
The screen appears scractched when examined carefully. (WITHOUT a comma before when)
You can definitely place it at the end.
To some people’s ears, it might sound better at the beginning. Certainly having it at the beginning changes the emphasis. But that does not change the grammatical correctness of doing it your way. As I wrote before, it’s fine.
No comma. 
This produces a modifier ambiguity. Is the screen scratched but its scratch is only visible upon close examination, or is the screen scratched as a result of close examination. Thus, the sentence as written above would not appear on the exam.
It is true that practice books often produce exercises that are not representative of the actual exam. It is also true that the exams themselves are less than perfect. From time to time, one finds multiple problems in an exercise and must decide which one the exam writer was more likely to have written intentionally, and which one crept in accidentally. Language Log has expounded upon this several times.