Entering Paradise before praying a single prayer to Allah

It is narrated from Sayyiduna Abu Hurayrah (may Allah be pleased with him) that Amr bin Uqaysh (may Allah be pleased with him) had some usury (riba) from the Days of Ignorance still owed (to him by people), so he was reluctant to accept Islam until he had first collected the usury. When the day of the Battle of Uhud came, he asked: "Where are my cousins?" People said: "At Uhud." He asked: "Where is so-and-so?" They said: "At Uhud." Then he (at once) put on his armour, mounted his horse and headed towards Uhud. When the Muslims saw him, they said: "Keep away from us, O Amr (for you are not a Muslim)." He said: "I have believed." Then he fought until he was wounded, and he was carried to his family wounded. Sayyiduna Sa'd bin Mu'adh (may Allah be pleased with him) came to him and said to his sister: "Ask him whether he fought out of zeal for his people or for the sake of Allah and His Messenger." He replied: "Rather, for the sake of Allah and His Messenger." Then he died and entered Paradise, although he had not prayed a single prayer for Allah (after accepting Islam).
This hadith is sahih (authentic) on the condition of Imam Muslim, but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyiduna Sahl bin Sa'd (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: "Two supplications are not rejected (or he said: are seldom rejected): one at the time of the call to prayer (adhan), and the other at the time in battle when the two armies are locked together with one another." And in another narration there is the addition: "Also at the time of rain (the supplication is not rejected)."
This hadith has a sahih (authentic) chain of narration, but the two Shaykhs (Bukhari and Muslim) did not record it.