Are dogs and pigs najis (impure)?
Question

A Christian asked me why Islam regards dogs and pigs as impure, despite the fact that they are created by Allah. He is not asking about their flesh; rather he wants to know why Muslims are afraid to let a dog come near them lest it touch them. He wants to know in particular where this is mentioned in the Qur’an, or whether there is a hadeeth to indicate that. What are the other animals that are regarded as impure? I hope you can answer my question, because this person is interested in Islam.
Praise be to Allah.Firstly: 
Perhaps the reason why this Christian is confused is because he thinks that Allah, may He be exalted, cannot create anything evil or filthy or impure, but this is a mistaken belief, because Allah, may He be exalted, is the Creator of all things, good and evil, pure and impure, that which is clean and that which is unclean. Indeed, Allah, may He be exalted, created Iblees (Satan), who is the most evil of all creatures. 
There is divine wisdom in everything that He has created, by means of which life is sustained, and people are tested. 
See the answer to question no. 12558 for a response to a question like yours. 
In the answer to question no. 1239 you will find a discussion of the wisdom behind the creation of harmful animals. 
Secondly: 
The view that pigs and dogs are impure is not a ruling on which there is consensus. Rather it is the view of most of the scholars; one of the scholars who think that they are pure is Imam Maalik, and Shaykh al-Islam Ibn Taymiyah regarded this as more likely to be correct. Some of the scholars differentiated between dogs and pigs, saying that dogs are impure but pigs are not. This is the view of some of the Shaafa‘is. Abu Ishaaq ash-Shiraazi (may Allah have mercy on him) said: With regard to pigs, they are impure because they are worse than dogs, and because it is recommended to kill them even if they are not harmful, and it is stated in the texts that they are haraam. If dogs are impure, then pigs are even more so.  End quote. 
An-Nawawi (may Allah have mercy on him) said, commenting on the words of ash-Shiraazi quoted above: In his book al-Ijmaa‘, Ibn al-Mundhir narrated that there was scholarly consensus on the impurity of pigs, and this is the strongest evidence, if it is proven that there is consensus. But the view of Maalik is that pigs are pure so long as they are alive. With regard to the argument presented by the author [i.e., ash-Shiraazi], the same argument was presented by others without evidence. We do not have any clear evidence to suggest that pigs are impure when they are alive.
End quote from al-Majmoo‘, 2/568 
On the contrary, others said that dogs are pure but pigs are not. This is the view of some of the Hanafis. 
Al-Kaasaani al-Hanafi (may Allah have mercy on him) said: With regard to dogs, the discussion concerning them focuses on whether they are impure in and of themselves or not. Our shaykhs differed concerning that. Those who said that they are impure in and of themselves put them in the same category as pigs, thus the ruling on them is the same as the ruling on pigs. Those who said that they are not impure in and of themselves regard them as being like all other animals apart from pigs, and this is the correct view.
End quote from Badaa’i  as-Sanaa’i‘, 1/63 
See also the answer to question no. 69840, in which we have quoted from Shaykh al-Islam Ibn Taymiyah concerning the basic principle that all things – including animals – are pure; this includes a discussion on the idea that dogs are pure in and of themselves, and that it is only the saliva that is impure, because there is a text to prove that. 
With regard to the Muslims keeping away from dogs, that is for fear of being licked by the dog, because dogs usually stick their tongues out and lick things, and impurity caused by a dog is severe and can only be purified by washing seven times, one of which should be with dust, because the Prophet (blessings and peace of Allah be upon him) said: “The purification of the vessel of one of you, if a dog licks it, is to wash it seven times, the first time with dust.”
Narrated by Muslim, 279 
This is why Muslims keep away from dogs and are worried in case they lick them, because they want to keep themselves and their clothes clean. 
And Allah knows best.