I live in Australia and I keep a dog. Some Muslims brothers said that it is not allowable to keep the dog in one's home because the mercy of God does not descend into that house. They quote the ḥadīth to support their claim. Please tell me is it true and can we keep a dog at home. Pleases note that the dog stays only at the backyard of the house.
Your friends are right in saying that there are Prophetic traditions which tell us that the angels do not enter the house in which there is a dog. However, there are equally authentic aḥādīth which tell us that dogs can be kept. Consider the following narrative.
عن عدي بن حاتم قال قلت يا رسول الله صلي الله عليه و سلم إني أرسل الكلاب المعلمة فيمسكن علي وأذكر اسم الله عليه فقال إذا أرسلت كلبك المعلم وذكرت اسم الله عليه فكل قلت وإن قتلن قال وإن قتلن ما لم يشركها كلب ليس معها
'Aadī b. Hātim narrated: I spoke to the Holy Prophet (sws) saying that I send forth the hounds and they kill a prey and keep it for me. And I invoke the name of God on it. [Can I eat the flesh of such a prey?] The Prophet (sws) replied: If you send your hound and pronounce Allah's name on it [while dispatching it] then you can eat the flesh of the prey. I asked: Even if the hound kills the prey? [He said]: Yes, unless the prey has not been killed by the prey with the help of another dog which was not sent with them [by you], the Prophet (sws) answered. (Muslim No: 1929)
Now we see that this narrative implies we can even eat the flesh of an animal killed by the hounds. This obviously implies that dogs can be kept and used as hounds. This is also confirmed by the Qur'an. Consider the following verse:
They ask you what is allowed them as lawful [to eat]. Answer, such things as are pure (tayyibāt) are allowed you; and what you teach animals of prey to catch, training them up though the skill which God has taught you. Eat, therefore, of that which they shall keep for you; and pronounce the name of God thereon; and fear God, for God is swift in taking an account. (Q 5:4)
يَسْأَلُونَكَ مَاذَا أُحِلَّ لَهُمْ قُلْ أُحِلَّ لَكُمُ الطَّيِّبَاتُ وَمَا عَلَّمْتُم مِّنَ الْجَوَارِحِ مُكَلِّبِينَ تُعَلِّمُونَهُنَّ مِمَّا عَلَّمَكُمُ اللّهُ فَكُلُواْ مِمَّا أَمْسَكْنَ عَلَيْكُمْ وَاذْكُرُواْ اسْمَ اللّهِ عَلَيْهِ وَاتَّقُواْ اللّهَ إِنَّ اللّهَ سَرِيعُ الْحِسَابِ
The above mentioned narrative indeed explains this verse. We also know that God Almighty mentions the Companions of al-Kahf (Q 18) who remained in the cave for a long time. God says they had a dog with them. This dog did not cause a break in the supply of God's mercy to them. For they were continually been fed by God.
In the light of these facts we can conclude that one can keep dogs. This is expressly approved by God in the Holy Qur'an.