Are dogs pure or impure? {Say: Lawful for you are [all] good foods and [game caught by] what you have trained of hunting animals which you train as Allah has taught you.} [Quran 5:4]
Ibn Mughaffal reported that the Messenger of Allah, sallallaahu 'alayhi wa sallam, ordered killing the dogs and then said, “What about them?” (i. e. about other dogs) and then granted concession (to keep) the dog for hunting and the dog for (the security) of the herd, and said, “When the dog licks the utensil, wash it seven times, and rub it with earth the eighth time.” [Sahih Muslim]
So in the Quran, it is mentioned that you can eat what your hunting animals have brought for you, but in the hadith it says that if a dog licks the utensil, we need to wash it seven times. Is this a contradiction? Another question is: why would the Messenger of Allaah, sallallaahu 'alayhi wa sallam, order us to kill dogs given that they were created by Allaah alone?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.

The majority of the scholars are of the view that the dog is impure, and they provided evidence via the hadeeth that you mentioned in the question about a dog licking a utensil, whereas the Hanafi scholars are of the view that the saliva of the dog is impure while its hair is pure. The Maalikis and some other scholars are of the view that dogs [their saliva and hair] are pure.

An-Nawawi favored the view of the majority of the scholars, as he said:

“Our view is that all of the dog is impure, whether it is trained or not, and whether it is small or big; this is the view of Al-Awzaa-i, Abu Haneefah, Ahmad, Is-haaq, Abu Thawr and Abu ‘Ubayd.

Az-Zuhri, Maalik and Daawood said, 'The dog is pure, but the utensil that the dog licked must be washed as an act of worship, [and not as a way of purifying it].' This was reported from Al-Hasan Al-Basri and `Urwah ibn Az-Zubayr, and they provided proof for their view with the saying of Allaah (which means): {So eat of what they catch for you} [Quran 5:4]; Allaah did not mention washing the spot where the dog caught the animal. They also provided evidence with the hadeeth of Ibn ‘Umar that says that 'Dogs used to pass inside the mosque at the time of the Prophet and they did not sprinkle anything over it (i.e. the Companions did not sprinkle water over the places where the dogs passed).' This was reported by Al-Bukhari in his Saheeh... Al-Bayhaqi reported this hadeeth with a continuous chain of narration and then said, 'The dogs used to pass inside the mosque and they did not sprinkle anything over it.' The scholars of our School (Shaafi'i) provided evidence with the hadeeth by Abu Hurayrah that  says that the Prophet said, 'If a dog licked one of your utensils, then spill what is inside it and then wash it seven times.' [Muslim] Abu Hurayrah also narrated that the Prophet said, 'To purify your utensil that a dog had licked, wash it seven times, the first time of which with soil.' [Muslim] Another narration reads, 'The purification of your utensil if a dog licked it is to wash it seven times.' [Muslim] The significance of the first hadeeth is clear, because if dogs were not impure, then the Prophet would not have commanded to spill what is inside it, as in that case it would be wasting (food or water) whereas we were forbidden to waste. The significance of the second hadeeth is also clear because purification is needed after one has invalidated his state of ablution or because of impurity; it is indeed not possible to interpret this (hadeeth) by not being in a state of ablution, and it is only possible to interpret it by purifying from an impurity. The scholars of our School answered their use of the verse (which means: So eat of what they catch for you. [Quran 5:4]); as evidence by saying that we have a known difference of opinion (with them) about whether or not it is an obligation to wash anything touched by a dog, so if we do not make it an obligation to wash it, then it is excused for a need and for the hardship involved in washing it, contrary to a utensil [which is easy to wash]. As regards the answer to the hadeeth of Ibn 'Umar, then Al-Bayhaqi while answering this, said, 'All the Muslims agreed in consensus that the urine of a dog is impure and that it is an obligation to sprinkle water over the urine of a child, so the impurity of a dog is more evident ... so the hadeeth of Ibn 'Umar was before the command to wash (a utensil) from the licking of a dog or (it is applicable) when the place of urine is not known, but if one is sure about it [the place of urine], then he must wash it; and Allaah Knows best.'”

Shaykh Ibn Taymiyyah favored the view of the Hanafi School, and he mentioned some pieces of evidence for this view being the preponderant one, as he said:

“Regarding the dog, there are three known views of the scholars:

1 – That the entire body of the dog is impure, even its hair. This is the view of Ash-Shaafi’i and one of the two views narrated from Ahmad.

2 – That the dog is pure, even its saliva. This is the view of Maalik in his well-known view.

3 – Its saliva is impure and its hair is pure. This is the view of Abu Haneefah that is well-known from him. This is the view that is supported by most of the scholars of his School; it is the second view of Imaam Ahmad and it is the most preponderant view in this regard.

So if one's clothes or body is touched by the wetness of the hair of the dog, then one has not become impure because of this, and if the dog drinks in water, then the water is spilled.”

In our view, the preponderant opinion in this issue is the view of the Hanafi School and Shaykh Ibn Taymiyyah .

On the other hand, there is no contradiction between the issue of the impurity of the dog and the issue of the permissibility of keeping it for hunting and guarding and the issue of killing some dogs, like the black one. There is a difference of opinion among the scholars about the impurity of the dog, as we have already mentioned. As regards the permissibility of keeping a dog for purposes such as hunting and guarding and the impermissibility of keeping it for other purposes, then it is confirmed that Ibn Al-Mughaffal said, “The Prophet ordered the killing of dogs, and then said, 'What is the trouble with them (the people of Medeenah)? How dogs are nuisance to them)? and then granted concession to keep the dog for hunting and for (guarding) the herd.” [Muslim]

Also, Abu Hurayrah narrated that the Prophet said, “Whoever keeps a dog, except a dog for hunting, herding livestock or guarding farms, will lose out of his good deeds equal to one Qiraat every day.” [Al-Bukhari, Muslim and others]

Besides, ‘Aa’ishah narrated that the Prophet said, “The angels do not enter a house in which there is a dog or an image.” [Al-Bukhari, Muslim and others]

An-Nawawi said, “The Prophet gave permission regarding hunting dogs and sheep dogs, and in another hadeeth, a farming dog, but he forbade keeping dogs other than these. The scholars of our School and others agreed that it is forbidden to keep a dog without a need, such as keeping a dog because of its appearance or for boasting with it; this is forbidden without any difference of opinion.”

With regard to the wisdom of killing dogs, the Prophet did not command killing all dogs; rather, he ordered the killing of a mordacious (biting) dog due to its harm. ‘Aa’ishah narrated that the Prophet said, “There are five harmful creatures that should be killed even if one is in the state of Ihraam (ritual state assumed for Hajj or 'Umrah): a snake, a speckled crow, a rat, a voracious dog, and a kite.” [Al-Bukhari and Muslim]

Moreover, the Prophet said, “If dogs were not a nation among nations, then I would have commanded that they be killed. So kill any all-black dog.” [Abu Daawood, at-Tirmithi, An-Nasaa’i and Ibn Maajah]

However, according to the most preponderant opinion, the command to kill the all-black dogs is abrogated as stated by An-Nawawi, “The scholars agreed about killing a voracious dog, but they differed about dogs that do not harm. Imaam al-Haramayn (al-Juwayni) said, 'The Prophet commanded the killing of all dogs and then this was abrogated with the exception of all-black dogs, and then the Sharee'ah established that it is prohibited to kill all kinds of dogs that do not harm, even the all-black dog.” 

Allaah knows best.