How sound is the story of the Prophet, may Allah’s prayers and peace be upon him, ascending to Paradise with Gabriel, upon him be peace, and Gabriel’s showing him the Prophet’s house therein, while being unable to enter it so long as he still has a lifespan in this world? And is this part of the journey of Isra and Mi'raj?

This incident is true, but it was a vision seen by the Prophet, peace and blessings be upon him, and not the well-known Isra' and Mi'raj. Al-Bukhari narrated this long vision, in which the Prophet, peace and blessings be upon him, saw the torment of liars, those who abandon the Qur'an, fornicators, and devourers of usury. He also saw Ibrahim, peace be upon him, Malik, the guardian of Hellfire, the abode of the believers, and the abode of the martyrs. He learned that the two men who accompanied him were Jibreel and Mika'eel, and they informed him that he has a great status in Paradise.