Does touching the private part invalidate wudoo? 

Removing impurity with water.

I wiped over my shoes & later took them off, can I wipe over my socks afterwards?

After marital relations you are in a state of ritual impurity (called Janaba where recitation of Qur'an is not allowed) which is different from the state of menstruation. During menstruation even though there is a difference of opinion you are allowed to recite Quran without touching it physically. Some ulema even encourage it. 

A vast majority of scholars prefer washing feet during Wudu. It is the asal and the basic principle. But let's also remember it is sunnah to wipe over socks. 

How to Make Wudu (Ablution) in the correct way.

Does the child's vommit nulify wudu?

Wiping over khuff during ablution is a form of Islam easiness when it is hard to remove it or its likes. But some conditions make it invalid like; time limit and feet exposure. More cases are clarified in this article.

There are some things that make ablution invalid; such as any discharge from private parts and this applies to stool, urine, semen, blood, and wind. Also, Deep sleep and touching one’s private parts break ablution.

Kuff is what is worn over the feet. It includes socks and could be made of various materials. God has permitted his servants to wipe over it during ablution if they cover the feet up to the ankles as a form of easiness. 

Wudu means cleanliness and brightness. But in Islam, it refers to washing the face, arms, head, and feet with clean water. In other words it is called ablution. It is obligatory before prayer or similar worship acts.

Wudu means cleanliness and brightness. But in Islam, it refers to washing the face, arms, head, and feet with clean water. In other words it is called ablution. It is obligatory before prayer or similar worship acts.

Of the unique actions of worship that Allah (subḥānahu wa ta'āla) has blessed us with in exclusion of all previous Ummahs before us is the ritual of Wudu (ablution).

In Arabic, the word Tayammum literally means an 'aim' or 'purpose.' In Islamic Law, it refers to: 'Aiming for or seeking soil to wipe one's face and hands with the intention of purification and preparing oneself to pray, and so on."