Is the Hadith: "Paradise is forbidden for the children of zina (fornication/adultery)" authentic, and what is their sin in this prohibition? And has a Hadith been narrated stating that Israel will occupy Palestine, then Iraq, Sham (Syria/Levant), and Egypt?

The aforementioned hadith (definition) regarding the child of zina (fornication/adultery) is fabricated and a lie. The child of zina does not bear the sin of his parents, as Allah Almighty says: "And no bearer of burdens will bear the burden of another," and "And that there is not for man except that for which he strives." He is born upon fitra (natural disposition) like other human beings.

As for the occupation of mentioned countries by the Jews, there is no evidence in the hadiths confirming this. There are hadiths from which it can be inferred that the Jews will not expand beyond their current state and will not occupy more than Palestine. The decisive battle will take place between them and the Muslims, among these is the hadith about fighting the Jews, in which stones and trees will say: "O Muslim, this is a Jew, come and kill him," and the hadith that the battle will take place at the Jordan River and that the Jews will not cross it.