How to Remove Dog's Impurity from Carpets
Question

How can we purify fitted carpets, like the carpet in the mosque or in a car, from the impurity or leftover water of a dog?
Praise be to Allah.How should a Muslim remove dog's impurity from carpets?
If the impurity gets onto this carpet that cannot be wrung out because it is too big or because it is attached to the ground or the car (fitted carpet), then the way to purify it is by removing the substance of the impurity that got onto it , and drying up whatever urine is on it, then pouring water over it and drying it up. This should be done several times until it is thought most likely that the impurity has been removed.
If the impurity came from a dog , then it is essential to wash the carpet seven times with water, as noted above, and to make the first time with soap or any other agent that will remove it; it does not have to be dust or soil.
Shaykh Ibn `Uthaymin (may Allah have mercy on him) was asked: What is the way to purify a large area rug of impurity? Is it important to squeeze it after removing the impurity itself?
He replied: 
“The way to clean a large area rug of impurity is to remove the impurity itself first of all, if it has any substance. So if it is solid, it may be picked up, and if it is liquid, like urine, it should be dried off with a sponge until it is removed. Then after that, water should be poured on it until it is thought most likely that the trace of it has been removed, or the impurity has been removed. In the case of urine, that may be achieved after doing that two or three times. As for squeezing it, that is not obligatory unless removal of the impurity depends on it, such as if the impurity got into the inside of the thing being washed, and it is not possible to clean its inside except by squeezing it. In that case it must be squeezed." (Fatawa Nur `ala Ad-Darb)
He (may Allah have mercy on him) was also asked: I heard on your program that the ground may be cleansed from the impurity of urine, if it has dried out, by the effect of the sun. Is it essential to have the effect of the sun, or [is it sufficient] just to let it dry out? Does that ruling apply also to carpets inside the house, whether they are fitted or not?
He replied: 
“When we say that the ground is purified by the sun and wind, that does not mean that it simply dries out. Rather it is essential to remove the traces [of the impurity] so that no trace is left of the urine or impure thing.
Based on that, we say: If urine gets on the ground and dries out, but the trace of the urine is still there, meaning that the trace of the spot remains, then it does not become pure by means of that. However, if some time passes, then its traces disappear, then it does become pure thereby, because impurity is a substance that must be removed. So if this substance disappears by any means, then it becomes pure.
With regard to carpets, the carpet with which the ground is covered must be washed; regardless of whether it is fitted or otherwise, it must be washed. The way to wash it is by pouring water on it, then drying it up with a sponge, then pouring water a second and third time, until it is thought most likely that the trace of impurity has been removed." (Fatawa Nur `ala Ad-Darb)
Do seats and places in which the dog sniffs and licks become impure?
Shaykh Ibn `Uthaymin(may Allah have mercy on him) was asked: There are some inspection points in some major companies that use trained dogs, which come into the front seat of the car then start to sniff and lick. Do the seats and places which the dog sniffs and licks become impure thereby?
He replied: 
“As for the sniffing, it does not matter, because no saliva comes out of the dog. As for licking, saliva does come out of the dog, and if the dog’s saliva gets onto clothing and the like, then it must be washed seven times. But we do not say that one of them should be with dust, because that may cause damage. Rather we say that instead of dust one may use soap or some other agent that will remove impurity, and that will be sufficient, along with washing it seven times." (Liqa’ Al-Bab Al-Maftuh  49/7)
There is no difference between the dog’s urine and excrement, and its saliva, according to the majority of jurists. Rather the urine and excrement are worse." (Ash-Sharh Al-Mumti‘,  1/417)
Is it permissible for Muslims to keep dogs?
It is not permissible to keep dogs except in the cases in which concessions are granted by the Lawgiver (Allah), because of the report narrated by Al-Bukhari (2145) from Abu Hurayrah (may Allah be pleased with him), who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever keeps a dog, one Qirat will be deducted from his good deeds every day, except a dog for farming or herding livestock .”
Muslim (2974) narrated from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever keeps a dog that is not a dog for hunting, herding livestock or guarding land , two Qirats will be deducted from his reward every day.”
And Allah knows best.