Before Prophethood

[Hani]

You said:

((41. “Our Lord! Forgive me and my parents, and (all) the believers on the Day when the reckoning will be established.”

Why would Nabi Ibrahim (a.s) pray for his idolatrous parents?))

Because he's not infallible as the Shia preach nor is there a concept of complete infallibility in Islam. Allah did not accept his Du`a' for his parents:

{And the request of forgiveness of Abraham for his father was only because of a promise he had made to him. But when it became apparent to Abraham that his father was an enemy to Allah, he disassociated himself from him. Indeed was Abraham compassionate and patient.}

In fact his Du`a' reads as follows:

{And forgive my father. Indeed, he has been of those astray.}

In other words, the man was a disbeliever.

As for our Prophet (saw) he never used to read a holy book before this one nor did he know about matters of religion as Allah says:

{And you did not recite before it any scripture, nor did you inscribe one with your right hand. Otherwise the falsifiers would have had [cause for] doubt.}

{And thus We have revealed to you an inspiration of Our command. You did not know what is the Book or [what is] faith, but We have made it a light by which We guide whom We will of Our servants. And indeed, [O Muhammad], you guide to a straight path}

So not much is known except that he used to go to the mountains and contemplate such as was mentioned in al-Bukhari:

[He (saw) used to go in seclusion in the cave of Hira where he used to worship continuously for many nights before going back to his family to take the necessary provision (of food) for the stay.]

However, his worship was no more than contemplation and reflection as we said previously, as he (saw) did not know of a specific ritualistic method of prayer nor did he consider himself to be a member of any certain faith.

As for the narration of the food, in the times of ignorance the pagans sacrificed for their idols and then invited people for the feast. Muhammad ibn `Abdillah (saw) was offered some meat but he refused to eat from it, it could be that his instinct prevented him from consuming from what was sacrificed for stone idols, so he (saw) passed it on to the person next to him but he also refused to eat from it (Zayd bin `Amro who was upon the faith of Hanifiyyah).

There is no issue here as there was no religious law prohibiting Muhammad ibn `Abdillah (saw) from offering the food to the next person in case he didn't want to eat it. More importantly, there was no religious law prohibiting him (saw) from eating from what his own tribesmen slaughtered or what other pagans slaughtered, so even if he ate and he had no other choice but to eat from what his own people ate, then that's not a problem at all especially since Islamically we are allowed to eat from the food of the Jews and the Christians and they're polytheists and disbelievers.

Whatever the details are and no matter what he (saw) was able to achieve from means of enlightenment before the revelation, they were incomplete and incomparable to those he (saw) acquired after the revelation:

{Did He not find you [O Muhammad] an orphan and give [you] refuge? - And He found you astray and guided [you], - And He found you poor and made [you] self-sufficient.}

NOTE: There is also a story of how the angels cleansed his heart (saw) when he was about ten years old and this is related. Link: http://islamqa.info/en/89869

[Khalifa]

You said:

((41. “Our Lord! Forgive me and my parents, and (all) the believers on the Day when the reckoning will be established.”

Why would Nabi Ibrahim (a.s) pray for his idolatrous parents?))

Because he's not infallible as the Shia preach nor is there a concept of complete infallibility in Islam. Allah did not accept his Du`a' for his parents:

{And the request of forgiveness of Abraham for his father was only because of a promise he had made to him. But when it became apparent to Abraham that his father was an enemy to Allah, he disassociated himself from him. Indeed was Abraham compassionate and patient.}

Are you talking about this promise? in Verse 47 of Surah Maryam.
 
 Conversation between Ibrahim A.S and His Father

My dear father, I fear lest a punishment from the All- Merciful (Allah) should afflict you, and you become a companion of Satan.” (45)

 He said, “Are you averse to my gods O Ibrahim? Should you not stop, I will definitely stone you. And leave me for good.” (46)

He said, “Peace on you. I shall seek forgiveness for you from my Lord. He has always been gracious to me. (47)