Who was the first Prophet after Adam (Alaihi As-Salaam)?While praying Qur'an if I clean the pamper and wash the stool can I pray Salah without ablution?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The first Prophet after Adam (Alaihi As-Salaam) was Nuh (Alaihi As-Salaam) who is also one of the Ulu al-'Azm (The possessors of constancy).
The evidence for this is the Hadith reported by al-Bukhari : "Then, they come to Nuh and say: 'O Nuh you are the first Messenger to the Dwellers of Earth and Allah called you a thankful servant …'" .
The answer to your second question is that the Wudu of the woman is not annulated by her cleaning her baby from impurities or touching his private parts.
This is the preponderant opinion of Muslim scholars. They say that what annulates Wudu is touching one's own private parts as proved by clear evidence. The reason the scholars believe touching private parts annulates the Wudu does not exist in the case of a woman touching her baby's private parts. Again it will constitute much hardship for the woman if asked to redo her Wudu anytime she touches her baby's private parts. Allah Says (interpretation of meaning): {…, and has not laid upon you in religion any hardship, …} [22: 78].
So, your Wudu remains valid even after washing your baby from impurities or touching his private parts.
Allah knows best.