All praise is due to Allah. May Allah's peace and blessings be upon the Messenger of Allah, his family and companions.
To proceed:
The issue of regarding the opinions of ibn 'Umar as evidence is a secondary issue from the issue of regarding the opinion of a Sahabi as evidence. The scholars of hadeeth, jurists, and experts in usool al-fiqh discussed this issue. Some of them wrote books dedicated to this topic. In short, there are different situations for the opinion of a Sahabi:
The first situation: If the opinion of a Sahabi is contrary to a text from Allah's Book and the Sunnah, then it will not be considered as evidence. Imam Shafi' (may Allah have mercy on him) stated in al-Umm 7/280, "As long as Allah's Book and the Sunnah are present anyone who hears evidence from them is not excused from following them. If that is not the case, we resort to the opinions of the companions of the Messenger of Allah (peace be upon him), or one of them." In Majmoo' al-Fataawa 1/284, Shaykh al-Islam said,"If the Sunnah is contrary to the opinion of the Sahaba, then the proof will be in the Sunnah of the Messenger of Allah (peace be upon him), and not in what contradicts it. This is the undoubted opinion of the scholars.
The second situation: If the opinion of the Sahabi was well-known, and none of the Sahaba rejected it, then it is a proof according to the majority of the scholars of hadeeth, the jurists, and the scholars of usool al-fiqh. In Majmoo' al-Fatawa 1/284, Shaykh al-Islam said, "If an opinion of the Sahabi was well-known and not opposed, then that was an affirmation. It may be said: "This is an affirmative consensus, because it will be known that the Sahaba affirmed this action, and none of them opposed it, and the Sahaba do not affirm an invalid action.
The third situation: If the opinion of the Sahabi was not well-known, and no other Sahaba knew about it, then the scholars have differed on this issue. The preponderant opinion is that such an opinion is a proof. This is the opinion of the majority of the scholars of hadeeth and jurists from the Hanafites, the Malikites, the Hambalites, and it is one of the opinions which were transmitted from Imam Shafi'.
The fourth situation: If the opinion of the Sahaba contradicted the views of the other Sahaba (may Allah be pleased with them), then the opinion of one of them is not a proof for the other Sahaba. This issue is a point of consensus among the scholars. Those who came after the Sahaba may look into the different opinions of the Sahaba to determine the preponderant opinion. However, this does not invalidate the usage of the opinions of the Sahaba as evidence because their difference of opinion is resolved by referring to the Qur'an and the Sunnah.
What was mentioned in your question with regards to the fact that ibn 'Umar used to lift his hands during takbeer in the Eid and funeral prayers vacillates between the first situation and the fourth situation.
It may be from the first situation because some of those who are of the view that hands should not be raised during takbeer in the Eid and funeral prayer used as evidence for their stance what was transmitted by ad-Daarqutni from the hadeeth of Abu Hurairah and ibn Abbas (may Allah be pleased with them) who narrated that when the Messenger of Allah (peace be upon him) performed the funeral prayer he would raise his hands in the first takbeer. Ibn 'Abbas added, "Then he would not raise his hands in takbeer for the rest of the prayer." Those who are of the view that the hands should be raised answered that these two hadeeths are weak.
The issue may be from the fourth situation because the Sahaba (may Allah be pleased with them) differed on this issue. Ibn Mas'ud and ibn 'Abbas held that the hands should not be raised except in the first takbeer, as was reported by ibn Hazm.
There is an issue which must be discussed and it concerns using the actions of ibn 'Umar as proof, and this is what the question asked about. The issue is that ibn 'Umar had ijtihaad on some issues, and he was contradicted by the senior Sahaba (may Allah be pleased with them) on those issues. Among those issues were the fact that ibn 'Umar used to put water into his eyes when making wudoo. He also was keen to move in the roads where the Prophet (peace be upon him) used to move. He would rest when the Prophet rested during his journeys, and he would make wudoo at the places that he saw the Prophet (peace be upon him) making wudoo. He would pour the water that remained from his wudoo on the tree that he saw the Prophet (peace be upon him) pouring and so on. These issues are part of the fourth situation, and the majority of the scholars were of the opinion that it was not recommended to follow ibn 'Umar in this regard based on the actions of the senior Sahaba who left these actions and did not do them. And Allah knows best.
