1- When Umar (RA) entered and saw the prophet with his wives, Scholars say that this was probably before the verse of Hijab has been revealed. Because the verse says:
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful.
AL-Ahzab : 59
And when you ask [his wives] for something, ask them from behind a partition. That is purer for your hearts and their hearts
AL-Ahzab : 53
2- Al-Nawawi concluded from this hadith that when a Muslim see his friend/brother sad or anxious he should try to cheer him up or make him laugh so that some of his sadness go away.
3- It is the human nature of a man/women to ask for more,so that is why the wives of the prophet wants more. and since muslims used to give him alot of money that he would spend it all on the poor and the needy. So they wanted to take some of that money and spend it on them. But the prophet refused and becames sad.
Neverthless, the prophet provided his wives with the basic and necessary things and he also asked muslims to do that in his last sermon said:
Remember that you have taken them as your wives only under Allah’s trust and with His permission. If they abide by your right then to them belongs the right to be fed and clothed in kindness. Do treat your women well and be kind to them for they are your partners and committed helpers. And it is your right that they do not make friends with any one of whom you do not approve, as well as never to be unchaste.
4- The prophet did not hit a women in his life and in another narration in
Musnad Imam ahmed the hadith says that the prophet asked Umar and Abu Baker to not hit their daughters. The narration but in Arabic
See this question for more info Does the Quran allow husbands punish their wives?
Here is a hadith that shows how the prophet used to deal with his wives in a relevant situation:
Nu’man ibn Basheer reported: Abu Baker came seeking permission to enter upon the Prophet, peace and blessings be upon him, and he heard Aisha raising her voice over the Messenger of Allah. He entered and said, “O daughter of Umm Ruman! Are you raising your voice over the Messenger of Allah?” So the Prophet stood between them and when Abu Bakr left, he said to her, “Do you see how I saved you from him?” Then Abu Bakr again sought permission to enter upon the Prophet and he heard them both laughing. He entered and said, “O Messenger of Allah, allow me to share in your peace just as you have allowed me to share in your conflict.”
Source: Musnad Ahmad 17927
5- As you can see that Umar(father of Hafsa) and Abu Baker(father of Aisha) disciplined their daughters because they asked for extra money from the prophet (PBUH) which he didn't had. From this hadith scholars concluded that the father has the right to discipline his son/daughter if that would make things better. and as a stubborn kid I must admit if it weren't for my dad disciplining me I would not have graduated from high school.
6- Imam Al-Nawai in his book "explanation of Sahih Muslim" ( 10 / 336 ) said: The word
يجأ means to slightly stab their necks. As to disciplined them and not to cause heavy damage to their necks. The Source is in Arabic