Assalaamu alaykum, there are some people who say things against Mu'aawiyah, may Allaa be pleased with him, because, according to them, he used to insult Companions, and they provide following evidences. 1. He insulted Abdullah bin Amr bin Al-'Aas when Abdullah narrated a hadith. He rejected his hadith and said, "Those men are the ignorant amongst you. Beware of such hopes as make the people go astray." Sahih al-Bukhari 3500 And Abdullah bin Amr was also worried about his sayings, so much so that Abd Rabb Al-Ka'ba said to Abdullah bin Amr bin Al-'Aas, "This cousin of yours, Mu'aawiyah, orders us to unjustly consume our wealth among ourselves and to kill one another, while Allah says: {O ye who believe, do not consume your wealth among yourselves unjustly, unless it be trade based on mutual agreement, and do not kill yourselves. Verily, God is Merciful to you." (iv. 29)." [Sahih Muslim] 2. He insulted 'Umar ibn Al-Khattaab and Ibn Umar; he said, "If anybody wants to say anything in this matter of the Caliphate, he should show up and not conceal himself, for we are more rightful to be a Caliph than he and his father." [Sahih al-Bukhari 4108] To this,Ibn Umar thought to respond by saying, "He who fought against you and against your father for the sake of Islam is more rightful to be a Caliph." [Ibid] 3. He insulted Ali: "Mu'aawiyah came on one of his pilgrimages, and Sa`d entered upon him. They mentioned `Ali, and Mu'aawiyah criticized him. Sa`d became angry and said, 'Are you saying this of a man of whom I heard the Messenger of Allah say, 'If I am a person's close friend, `Ali is also his close friend.' And I heard him say, 'You are to me like Harun was to Musa, may Allaah exalt their mention, except that there will be no Prophet after me.' And I heard him say, 'Today, I will give the banner to a man who loves Allah and His Messenger.''" [Ibn Maajah] Against that, these people say, "You have only historic narrations that are not proven against these authentic insults of Mu'aawiyah." Please reply.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

One of the tenets of Ahlus-Sunnah Wal-Jamaaʻah is maintaining pure hearts and tongues towards the honorable Companions of the Messenger of Allaah, sallallaahu ʻalayhi wa sallam, including Muʻaawiyah who used to write down what was revealed of the Noble Quran. Ahlus-Sunnah do not attribute infallibility to the Companions ; rather, they believe that the Companions may commit errors and sins. However, they do not take such errors and sins as justification for slandering them and undermining their refined status. On the contrary, they take them as their allies, love them, and try to find valid excuses for their mistakes.

The followers of religious innovations, who exaggerate in revering their imaams and claim their infallibility, assume that Ahlus-Sunnah hold the same corrupt belief for the Companions of the Prophet, sallallaahu ʻalayhi wa sallam, i.e. infallibility. Therefore, they strive in finding their faults, believing that by doing so they are attacking and embarrassing Ahlus-Sunnah. Verily, this is an outrageous mistake; Ahlus-Sunnah do not attribute infallibility to the Companions. Ahlus-Sunnah adopt a balanced approach in addressing the reports concerning the faults and mistakes of the Companions . Shaykhul-Islam Ibn Taymiyyah summed up the approach of Ahlus-Sunnah with the following precious words:

“They refrain from delving into the disputes that occurred between the Companions and say, 'These narrations that have been reported concerning their faults have in them some that are false, some that have had things added to them, some that have had parts omitted from them, and the original indication of some has been altered. As for those narrations that are authentic (concerning their faults), then they (the Companions) are excused (and forgiven) – they practiced ijtihaad (personal reasoning) and were either correct or wrong. Furthermore, they do not believe that each of the Companions is infallible and free from committing major or minor sins. Rather, they may commit sins in general, however, they possess from precedence and virtues that which necessitates that they be forgiven for whatever (sins) they commit – if they did commit any. This is even to the point that they will be forgiven for evil deeds the likes of which those who come after them will not be forgiven for. This is because they possess good deeds that wipe out the evil deeds, the likes of which will not be for those after them.'

It is established that the Prophet, sallallaahu ʻalayhi wa sallam, stated that they are the best of generations and that if one of them were to give a mudd (handful) of gold in charity, it would be better than if someone who came after them were to give the whole of Mount Uhud in gold. And if it was the case that one of them committed a sin, then either he repented from it, performed some good deed that wiped it out, or he will be forgiven for it due to the virtue of his precedence or because of the intercession of Muhammad, sallallaahu ʻalayhi wa sallam, of which the Companions are more deserving. Perhaps he was tested by some affliction in this worldly life, by which it was expiated from him. So if this is with respect to the sins that were actualized (by them), then what about the matters in which they did ijtihaad? If they were correct, then they will have two rewards, and if they were incorrect, then they will have just one reward and their error will be forgiven. Furthermore, the few bad deeds committed by some of them are slight in comparison to their virtues and good qualities, such as their belief in Allaah and His Messenger, their jihaad for His Cause, their migration, their support (of the Religion), as well as their beneficial knowledge and good deeds.” [Al-'Aqeedah Al-Waasitiyyah]

The following lines will provide some clarification about the reports mentioned in your question:

As for Muʻaawiyah's statement, “Those men are the ignorant amongst you,” he did not particularly name Ibn ʻUmar in the first place; he might have been referring to another person. Ibn Hajar wrote, “Muʻaawiyah might have been referring to other than ʻAbdullaah ibn ʻUmar in this report...” [Fat-h Al-Baari]

As for the statement about ʻAbd Ar-Rahmaan ibn ʻAbd Rabb Al-Kaʻbah, it does not imply any evidence to this effect nor is it related to what you have mentioned about Ibn ʻUmar's anger towards Muʻaawiyah .

As for Muʻaawiyah's remark, “We are more rightful to be a Caliph than he and his father,” it does not imply an insult for Ibn ʻUmar . Rather, it indicates that Muʻaawiyah believed himself worthier of the Caliphate because he held that the candidate who is more rightful to be a Caliph should be the one endowed with the greater share of strength, soundest judgment, most abundant knowledge, and not the one who has embraced Islam first and is endowed with a greater share of religiosity and devotion in worship. This is why he remarked that he was more rightful to be a Caliph than Ibn ʻUmar. However, Ibn ʻUmar held otherwise; he maintained that leadership of a less eligible candidate for the Caliphate is not accepted in case a more eligible one is available unless it would incur fitnah (dissension and conflicts).

As for Muʻaawiyah's remarks about ʻAli they could be intended to be a mere reference to a mistake committed by him or the like and not an insult or curse, as As-Sindi asserted in his Haashiyat As-Sindi ʻala Ibn Maajah, “The hadeeth that reads that 'He abused him' means that Muʻaawiyah offended ʻAli and insulted him and even commanded Saʻd to insult him as well, as cited by Muslim and At-Tirmithi . The reason for such an incident was the disputes that took place between them over the worldly affairs. We implore Allaah to forgive us all and pardon our sins. Out of having the required good expectations of our fellow Muslim, Muʻaawiyah's attitude may be interpreted as a personal view that  'Ali' was wrong and the like of the acceptable actions on part of those competent for practicing ijtihaad and not actual cursing.”

Allaah knows best.