Did the faith of the Prophets increase, and what is the meaning of the hadith, “We are more likely to have doubts than Ibraaheem did”?
Question

We believe prophets are sinless. 1. Did some incident that happened during their time increase their imaan?  2. Please explain the meaning of this ayaah? Quran 2:260. 3. What is the explanation of this hadith?
Praise be to Allah.Firstly: The faith of the Prophets increased, and never decreased
The faith of the Prophets increased, and never decreased.
Allah, may He be exalted, says (interpretation of the meaning):
{Indeed, We have given you, [O Muhammad], a clear conquest, 
That Allah may forgive for you what preceded of your sin and what will follow and complete His favor upon you and guide you to a straight path, 
And [that] Allah may aid you with a mighty victory.
It is He who sent down tranquillity into the hearts of the believers that they would increase in faith along with their [present] faith. And to Allah belong the soldiers of the heavens and the earth, and ever is Allah Knowing and Wise} [al-Fath 48:1-4].
Shaykh Ahmad ibn Hajar Aal Butaami (may Allah have mercy on him) said, after explaining that faith may increase and decrease:
That is applicable to those other than the Prophets and the angels, because the faith of the Prophets increases and never decreases, and we cannot say that that which may increase may also decrease, because we say that their faith is exempted because of infallibility. The faith of the angels neither increases nor decreases, because their faith is an inherent part of their nature, and whatever is like that cannot vary. Some of the scholars said that the angels are like the prophets, and their faith increases and does not decrease. And there is no evidence to prove otherwise." (Al-‘Aqaa’id as-Salafiyyah bi Adillatiha an-Naqliyyah wa’l-‘Aqliyyah, p. 425).
Shaykh ‘Abd al-Muhsin al-‘Abbaad (may Allah preserve him) was asked: Does the faith of the Prophets and Messengers increase and decrease, or does it increase only?
He replied: Undoubtedly they are the most perfect of humankind; they are perfect in all ways, and are not subject to imperfection. Rather their faith only increases.
End quote from http://iswy/co/e437s
Secondly: The Prophets are infallible and protected from doubt and disbelief
The Prophets are infallible and protected from doubt and disbelief, even before their missions begin.
Allah granted wisdom and sound judgement to Ibraaheem (peace be upon him) from an early age, as He says (interpretation of the meaning):
{And We had certainly given Abraham his sound judgement before, and We were of him well-Knowing} [al-Anbiya’ 21:51].
That is, before he reached adulthood, as was narrated from Ibn ‘Abbaas.
See: Zaad al-Maseer by Ibn al-Jawzi (3/194).
He (blessings and peace of Allah be upon him) never experienced any doubt; rather he only wanted to see how Allah would give life to the dead, so that he could move from certain knowledge to certain truth, because hearing is not like seeing.
Allah, may He be exalted, says (interpretation of the meaning):
{And [mention] when Abraham said, “My Lord, show me how You give life to the dead.” [Allah] said, “Have you not believed?” He said, “Yes, but [I ask] only that my heart may be satisfied.” [Allah] said, “Take four birds and commit them to yourself. Then [after slaughtering them] put on each hill a portion of them; then call them - they will come [flying] to you in haste. And know that Allah is Exalted in Might and Wise”} [al-Baqarah 2:260].
Ibn Katheer (may Allah have mercy on him) said in his Tafseer (1/689): The scholars mentioned several reasons for this request of Ibraaheem (peace be upon him), including the fact that when he said to Nimrood, {My Lord is the one who gives life and causes death } [al-Baqarah 2:258], he wanted to progress from certain knowledge concerning that to certain truth, and to see that with his own eyes. Hence he said: {“My Lord, show me how You give life to the dead.” [Allah] said, “Have you not believed?” He said, “Yes, but [I ask] only that my heart may be satisfied.”} [al-Baqarah 2:206]. End quote.
With regard to the words of the Prophet (blessings and peace of Allah be upon him), “We are more likely to have doubts than Ibraaheem did”, this is a negation of doubt on both their parts. in other words, I do not doubt, and if Ibraaheem doubted, then we are more likely to do that than him. This was an expression of humility on the part of the Prophet (blessings and peace of Allah be upon him).
The hadith is narrated by al-Bukhaari (4537) and Muslim (151) from Abu Hurayrah (may Allah be pleased with him), who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “We are more likely to have doubts than Ibraaheem did when he said: {“My Lord, show me how You give life to the dead.” [Allah] said, “Have you not believed?” He said, “Yes, but [I ask] only that my heart may be satisfied.”} [al-Baqarah 2:206].
An-Nawawi said in Sharh Muslim (2/183): The scholars differed concerning the meaning of the words “We are more likely to have doubts than Ibraaheem did”; the best and soundest of their views is that which was stated by Imam Abu Ibraaheem al-Muzani, the companion of ash-Shaafa‘i, and a number of other scholars, which is that what is meant is that doubt is impossible in the case of Ibraaheem, for if it is possible for Prophets to have doubts about the revival of the dead, then I would be more likely to have doubts about that then Ibraaheem, and you know that I never doubted, so you should realise that Ibraaheem (peace be upon him) never doubted either.
Ibraaheem (blessings and peace of Allah be upon him) was only singled out for mention because the verse may be misinterpreted by people with little understanding as meaning that it is possible for Prophets to have doubts. The Prophet (blessings and peace of Allah be upon him only said that he was more likely to have doubts than Ibraaheem out of humility and good manners, or he said that before he (blessings and peace of Allah be upon him) came to know that he is the best of the sons of Adam. End quote.
Al-Khattaabi (may Allah have mercy on him) said in Sharh al-Bukhaari (3/1545): The way to understand  this hadith is that is reflects humility and avoidance of self-praise.
There is nothing in his words “We are more likely to have doubts than Ibraaheem did” to suggest that he was admitting that he or Ibraaheem (peace be upon him) had any doubts; rather it is a statement that neither of them had doubts at all. Hence we may understand from it that he meant the following: I do not have any doubts myself, and I do not doubt the power of Allah, may He be exalted, to give life to the dead, so it is more appropriate to say that Ibraaheem had no doubts about that.
From this hadith we may understand that Ibraaheem’s request of his Lord was not motivated by doubt; rather it was by way of asking to increase his knowledge about this fact (revival of the dead) and to see with his own eyes how Allah would bring that about. A person may feel more assured when he learns how a thing will be done than merely hearing a statement about it. In either case, knowledge of the matter is attained and doubts are erased.
And Allah knows best.