1) It is clear to me now that the use of the subjunctive "realized" is correct, but regarding the version with "realize", is it also correct here or is it definitely wrong?
It's not wrong. You can use 'realize'. What is wrong is saying that 'realized' in incorrect. As I say, sometimes you have a choice and, in some cases like this, either the indicative or subjunctive form are acceptable.
2) When one uses, in the original sentence, "realized" instead of "realize", how can you tell that the first form is subjunctive while the second is not? For instance:
"What if he goes to the shop?" and "What if he went to the shop?" - aren't both forms in the subjunctive irrespective that the first is simple present while the latter simple past?
"What if he go to the shop" is not a possible sentence, because the appropriate subjunctive form is "went" not "go".
"What if I were rich" NOT *"What if I be rich".
The two situations are both hypothetical, but only the second ("What if he went to the shop?") is in the subjunctive form.
3) In your examples above what is exactly the subtle difference? I still can't get all of it.
Good question. I don't have an easy answer. Probably using "What if I go ..." implies more willingness to go than "What if I went ..."
A: Ack! We're out of chocolate!
B: What if I go to the shop and buy some?
A: That would be great.
C: I want to watch channel X on TV.
D: I want to watch channel Y!
C: What if I went to the shop and bought you some chocolate?
D: Then I'd let you watch channel X.
The difference could be explained in other ways in other contexts. Maybe someone else has an idea.