There is no doubt that the prophets are infallible and do not commit sins that will be subject to godly wrath and punishment. So what the holy Quran stated - about the eating of Adam from the tree - is that the divine order to refrain from such an act is not an order that requires obedience because of the relation of servitude between Almighty Allah and His slaves, but as an advisory suggestion that eating from the tree will cause the harm of being departed from Heaven.

What is popularly known is that Luqman was granted wisdom and was not a prophet, as this is also mentioned in some narrations. And Allah knows best.

In the Quran and authentic narrations, we find that prophets and their vicegerents suspended the usual laws of nature, like when Jesus cured the blind and the lepers, and when the vicegerent of Prophet Solomon brought the throne of Queen Belqees faster than the twinkling of an eye.
There is no doubt that similar acts took place by our Prophet and the Imams (peace be upon them) as many authentic narrations indicate.

This topic entails a detailed discussion, but in summary Allah said: “And if, when they had wronged themselves, they had but come unto thee and asked forgiveness of Allah, and asked forgiveness of the messenger, they would have found Allah Forgiving, Merciful” (4:64). It is clear that the intercession of the Prophet (peace be upon him and his progeny) is one of the causes of divine mercy and forgiveness, and there is no evidence that this is limited to his life, especially with the traditions narrated from him that indicate that he replies to our salutations and greetings to him after his death; so just as salutations reach him after his death, so do our requests for his intercession.

Intercession means to seek from the prophets and divinely appointed holy leaders the mercy of Allah and His forgiveness. It is a Quranic concept confirmed by the teachings of the Prophet and the Ahlulbait (peace be upon them all). However, a righteous devotee is encouraged to avoid sins, and if they occur, to repent to Almighty Allah and return to Him asking forgiveness and mercy. At the same time, he should ask the Prophet and the Ahlulbait (peace be upon them) to intercede and ask the Almighty Allah to grant him His mercy and acceptance. There is no doubt that joining between repentance and the intercession is more secure to obtain salvation rather than relying on repentance alone.
Having said that, this does not mean that one should rely on intercession and commit sins as this leads to disregarding sins, which is one of the greatest sins that may disqualify the person from deserving intercession. Committing sins while belittling them leads to disregarding Allah's punishment and it is amongst the greatest sins. This might lead to losing the good end of this life and not dying as a believer, which is - no doubt - one of the conditions of deserving intercession. There is a difference between the desire of getting the intercession and the hope that one gets it, on one hand, and depending on intercession and committing sins because of this dependence on the other hand. The former places the person far from committing sins, whereas the latter encourages the person to commit sins and may lose, at the end, the chance to get the intercession.