Can You Wipe Over Shoes?
Question

Some people in America and Canada wear socks made of cotton or wool that go up to the knees, and wear shoes over them, but the shoes do not reach the ankles. Is it permissible to wipe over these shoes when doing wudu? If they take off their shoes, is their wudu still valid? When they go to pray, they take off their shoes; is their wudu still valid?
Praise be to Allah.Wiping over shoes
If the shoe covers the foot with the ankles, then it is permissible to wipe over it, because it is like the khuff [a type of leather sock that covers the foot and ankle; the word may also refer to a boot or any footwear that also covers the ankle].
But if it does not cover the area that must be washed, which is the entire foot with the ankle, then it is not permissible to wipe over the shoes according to the majority of jurists. (See: al-Mawsu‘ah al-Fiqhiyyah al-Kuwaitiyyah, 27/264)
This is the view favoured by Shaykh Ibn Baz and the Standing Committee for Issuing Fatwas.
Shaykh Ibn Baz (may Allah have mercy on him) said: “One of the conditions of it being permissible to wipe over the khuffayn and the socks is that they should cover the area that must be washed.” (Majmu‘ al-Fatawa by Ibn Baz, 10/111) See also: Fatawa al-Lajnah ad-Daimah, 5/396)
If you wipe over the shoes and take them off, is wudu valid? 
If a person wipes over the shoes that cover the area that must be washed, then he takes them off when he is still in a state of purity, his wudu is not rendered invalid, according to the correct scholarly view.
This issue has been discussed previously in the answers to questions no. 100112  and 26343 .
But it should be noted that the concession allowing him to wipe over them ends when he takes them off, so if he puts them on again and wants to do wudu, he has to take off his shoes and socks and wash his feet.
Scenarios of putting on socks and putting over them shoes that do not cover the ankles 
If he puts on socks and puts over them shoes that do not cover the ankles, then there are three scenarios:

He wipes over the shoe only; we have stated above that this is not permissible.
He wipes over the socks only, in the sense that he takes off his shoes and wipes over his socks with his hands, then puts the shoes back on. This is permissible and there is nothing wrong with it, and in this case it is permissible for him to take off the shoes, and that does not render his wudu invalid.
He wipes over the shoes and socks together; this is also permissible

If he wipes over the shoe that does not cover the ankle, and completes the wiping over the socks, then the ruling applies to both of them together.
If he takes off the shoe only, or takes it off with the socks , that does not render his wudu invalid and it is permissible for him to pray, but it is not permissible for him to wipe over them again after that, unless he has done wudu in full and washed his feet.
It says in Fatawa al-Lajnah ad-Daimah, 5/396):
“The one who does wudu may wipe over his sock only, or wipe over his shoe only, if it covers the ankles and the skin of the feet cannot be seen beneath it.
If it does not cover the ankles, he may wipe over it if it is worn over socks that do cover the ankles, and also wipe over what appears of the socks above the area that is to be washed in wudu, and he may pray wearing both of them.”
Shaykh Ibn Baz (may Allah have mercy on him) said: 
“With regard to the shoe, it is like the sandal: if it does not cover the foot plus the ankles, then if he wipes over the shoes with the socks, the ruling applies to both, and if he limits it to wiping over the socks only, that is sufficient. In that case it is permissible for him to take off the shoes whenever he wants, and his wudu remains valid, because the ruling of wiping is connected to the socks only.” (Majmu‘ Fatawa Ibn Baz, 29/73)
We would like to point out that the rulings having to do with the khuffayn are applicable to socks and shoes/boots that cover (the feet plus the ankle), because the ruling on both is the same, according to the more correct view.
And Allah knows best.