Did the Prophet Solomon (peace be upon him) have intercourse with one hundred, ninety, sixty, or seventy women in a single night? And is this possible chronologically and health-wise? And why did he not say "Insha'Allah" (God willing) despite being a prophet? And what is the meaning of "شق رجل" (a part of a man/half a man) in the mentioned hadiths?

Regarding your first question, Allah's Prophet Sulayman (peace be upon him) had intercourse with over ninety and under one hundred women that night, sixty of whom were free women and the rest concubines. This number is not impossible rationally, for Allah is capable of granting strength for intercourse beyond the usual, and prophets have been granted special strength. As for Sulayman's not saying "Insha'Allah" (if Allah wills), it was due to forgetfulness, and this does not diminish his standing as a noble prophet, for reliance on Allah was firmly established in his heart, though he did not utter it. Finally, the meaning of "half a man" (شق رجل) is his half.