Raising the hands with all four takbirs in the funeral prayer:
حَدَّثَنَا وَكِيعٌ، عَنْ سُفْيَانَ، عَنْ عَاصِمِ بْنِ كُلَيْبٍ، عَنْ أَبِيهِ، أَنَّ ابْنَ عُمَرَ كَانَ يُكَبِّرُ عَلَى الْجَنَازَةِ، فَيَرْفَعُ يَدَيْهِ فِي كُلِّ تَكْبِيرَةٍ.
It is narrated that:
Waki‘ reported from Sufyan, from ‘Asim ibn Kulayb, from his father, that ‘Abdullah ibn ‘Umar (may Allah be pleased with them both) used to perform the funeral prayer, and he would raise his hands with every takbir.
(Al-Musannaf by Ibn Abi Shaybah, vol. 3, p. 256, hadith 1766)
There is no authentic hadith that explicitly mentions whether the Prophet ﷺ raised his hands or did not raise them in all four takbirs of the funeral prayer. However, the practice of the Companions is a valid example to follow, and Ibn ‘Umar (may Allah be pleased with him) used to raise his hands with each of the four takbirs in the funeral prayer.
The same general supplication can also be recited for a child, since supplication in the funeral prayer is not restricted, though there is a specific supplication for a child which is preferable.
عَنْ أَبِي هُرَيْرَةَ، قَالَ: كَانَ رَسُولُ اللَّهِ ﷺ يُصَلِّي عَلَى الصَّبِيِّ، فَيَقُولُ: «اللَّهُمَّ اجْعَلْهُ لَنَا فَرَطًا، وَسَلَفًا، وَذُخْرًا، وَشَفِيعًا مُجَابًا. اللَّهُمَّ أَعْظِمْ بِهِ أُجُورَنَا، وَثَقِّلْ بِهِ مَوَازِينَنَا، وَأَلْحِقْهُ بِصَالِحِ الْمُؤْمِنِينَ، وَاجْعَلْهُ فِي كَفَالَةِ إِبْرَاهِيمَ، وَقِهِ بِرَحْمَتِكَ عَذَابَ الْجَحِيمِ.»
It is narrated from Abu Hurayrah (may Allah be pleased with him):
The Messenger of Allah ﷺ used to pray over a child and say:
“O Allah! Make him for us a forerunner, a treasure, and an accepted intercessor. O Allah! Magnify our reward through him, make our scales heavy because of him, join him with the righteous believers, place him under the care of Ibrahim (peace be upon him), and by Your mercy, protect him from the torment of Hellfire.” (Sunan Ibn Majah, Book of Funerals, Hadith 1516; Musnad Ahmad, narration of Abu Hurayrah)