My question is: is wiping over both khuffayn at the same time obligatory and necessary, or is it what is preferable? Is it essential to wipe the right foot with the right hand and the left foot with the left hand? Please note that I am at work and cannot stand on the ground in my socks because the ground is dirty. So I wipe over each foot on its own, meaning that I stand on my left foot and wipe the right foot, and so on.
It is Sunnah when wiping the khuffayn to wipe both of them together, but that is not obligatory.
Some scholars are of the view that the right foot should be done first, but the correct view is the first one mentioned.
Al-Mardaawi said in al-Insaaf (1/185): The Sunnah way of wiping (over the khuffayn) is to put the hands with the fingers outspread on the toes, then draw the hands up towards the shins once on the right and left feet. It says in al-Talkhees wal-Bùlghah: It is Sunnah to give precedence to the right foot. Al-Bayhaqi narrated that the Prophet (blessings and peace of Allah be upon him) wiped over his khuffayn in one go, “it is as if I can see his fingers on the khuffaayn.” The apparent meaning of this is that he did not give precedence to one of them over the other. But however one wipes them it is acceptable. End quote.
From the words “however one wipes them it is acceptable” it may be understood that what you mention about wiping the right foot first and then the left does not matter; rather it is only different from what is preferable if one is able to do it.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: … i.e., what should be wiped is the top of the khuff, so he should pass his hand from the toes up to the shin only, and the wiping should be done with both hands on both feet at the same time, i.e., the right hand should wipe the right foot and the left hand should wipe the left foot at the same time, as the ears are wiped, because this is the apparent meaning of the Sunnah. Al-Mugheerah ibn Shu‘bah (may Allah be pleased with him) said: Then he wiped over them, and he did not say that he started with the right foot, rather he said that he wiped over them. So this is the apparent meaning of the Sunnah. Yes, if we assume that if he cannot use one of his hands, then he should start with the right foot before the left foot. But many people wipe with both hands on the right foot and both hands on the left foot; there is no basis for this as far as I know. … But however one wipes the top of the khuff is acceptable, but what we are talking about here is what is preferable.
End quote from Fataawa al-Mar’ah al-Muslimah, 1/250
It does not matter either if one wipes the right foot with the left hand, but the Sunnah is to wipe the right foot with the right hand and the left foot with the left hand, unless there is something wrong with one of the hands that means it cannot be used.
It says in Kashshaaf al-Qinaa‘ (1/119): It is Sunnah to wipe the right foot with the right hand and the left foot with the left hand, because of the hadeeth of al-Mugheerah quoted above. End quote.
And Allah knows best.