Is the narration that mentions that the Prophet (peace and blessings of Allah be upon him) offered his cloak to whoever prayed without thinking, and that Ali prayed and then told the Prophet that he had thought about which of the two cloaks he would give him, authentic? And what is its source?

The alleged hadith stating that the Prophet, peace and blessings be upon him, used to give money to the Companions in exchange for khushu' (devotion/humility) in prayer has not been found in any of the Sunnah books. It is well-known that he encouraged them to achieve khushu' by mentioning the great reward for it. It is reported in Al-Bukhari and Muslim that Uthman, may Allah be pleased with him, narrated that the Prophet, peace and blessings be upon him, said: «Whoever performs ablution like this ablution of mine, and then stands and prays two rak'ahs in which he does not speak to himself, his past sins will be forgiven.» A Muslim must verify any hadith attributed to the Prophet, peace and blessings be upon him, before disseminating it.