Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear questioner, we would like to thank you for the great confidence you place in us, and we implore Allah Almighty to help us serve His cause and render our work for His Sake.
First of all, we would like you to know that it is not allowed to keep a dog at home for a purpose other than hunting or guarding the house (watchdogs). The Prophet (peace and blessings be upon him) is reported to have said: "Whoever keeps a dog save for hunting or for guarding crops or cattle will lose one large measure (qirat) of his reward each day" (Al-Bukhari and Muslim).
In his response to your question, Sheikh `Abdul-Bari Az-Zamzami, a member of the Moroccan Scholars’ Association, states:
“Touching a dog does not nullify ablution because acts that nullify ablution are well known and touching a dog is not one of them. It should be noted that living animals such as dogs are not impure. Therefore, there is nothing wrong with offering Prayer in a place where dogs are roaming or passing, as Al-Bukhari reported in his Sahih that dogs used to enter the Mosque of the Prophet (peace and blessings be upon him) and then come out.”
For more information, you can read:
Acts that Nullify Ablution
A Dog Touched My Clothes: How to Cleanse Them
Muslims Owning Dogs: Permissible?
Keeping Dogs: Science Supports Shari`ah