Is it obligatory to inform someone who wishes to embrace Islam that the ruling on apostasy will be applied to them if they later abandon the faith? And what is the ruling on someone who initially claims to be a Christian to alleviate the fears of the addressed person, then clarifies their Islam?

There is no obligation to tell a person who wishes to embrace Islam everything at once. Rather, they should be taught the fundamentals of faith, the pillars of Islam, and its virtues gradually and in a manner appropriate to their situation. A scholar should educate people with knowledge progressively, starting with the smaller matters of knowledge before the larger ones. This is precisely what the Prophet, peace and blessings be upon him, did with Mu'adh ibn Jabal when he sent him to Yemen, instructing him to begin by inviting people to the Oneness of God, then to prayer, and then to zakat. This demonstrates the principle of gradual progression in da'wah (invitation to Islam), moving from the most important to the important.