If it's "the" flag, why isn't there a ״ה״ prefix? i.e. shouldn't it be "אני אוהב את הדגל הישראל?״" Or if it's construct state "דגל הישראל"
It's a construct, but Israel doesn't need the definite article, it's already definite without it. Generally names are considered definite without an article and don't take the article.
Another question: why is not necessary to insert "של", like "דגל של ישראל", "flag OF Israel"?
As mentioned above, it's in construct form; you don't need the 'של'. Like with עוגת שוקולד, you don't need the 'של', even though it is a cake /of/ chocolate.
Why isn't accepting "I like the Israel's flag" ? Is this answer wrong?
That is incorrect grammar in English. 'Israel' does not take a 'the'. As the commentator above said, it's already definite. (Similarly you wouldn't say 'the New York' or 'the Sally'.)