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.
Thank you for your question and interest to learn the teachings of Islam.
A Muslim should be careful to clean his body and clothes so as to be eligible to perform Prayers and other rituals of Islam. Purification and cleanliness are very important in Islam.
Responding to your question, Dr. Wael Shihab, PhD in Islamic Studies from Al-Azhar University, and currently the Imam of the Downtown Toronto Masjid in Canada, stated,
Thank you for your question and concern about purification which is an essential prerequisite for Prayer and other acts of worship. Abu Malik said:
The Messenger of Allah (peace and blessings of Allah be on him) said, “Purification is half of faith”. (Muslim)
Moreover, a Muslim family should not keep a dog without an acceptable reason. It is not allowed for a Muslim 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“. (Muslim)
So, you are, first of all, to advise your family not to keep a dog unless there is a necessity for keeping it at home. If there is an acceptable reason to keep a dog, then they are recommended to keep it in a particular place outside the house itself—maybe in the house’s garden—to avoid its impurities.
If they insist on keeping the dog, then you should take precautionary measures to prevent the dog from entering your own room and touching your clothes, cups, glasses, utensils, etc. These measures are meant to preserve your room, clothes, etc., clean. You should have a pure place to offer Prayers on.
As for washing your clothes in the washing machine that your family members use, there is nothing wrong in this. Don’t worry, your clothes will be clean and pure as long as you don’t see visible impurities on them.
Here, I’d like to tell you that there are differences among scholars regarding the impurities of dogs. One group of scholars, who may be deemed as a minority, consider dogs as wholly impure and unclean, another considers only their saliva as impure, while a third group considers dogs as wholly pure. In spite of their differences on this issue, they all agree that if a dog licks a container of water, it should be poured out, and then the container should be washed seven times, the first of which should be with mud.
Given the above, don’t worry about the purity of your clothes when you use the family washing machine. And, you are to avoid direct contact with the dog, particularly its salvia.
May Allah Almighty strengthen your and our faith and guide us all to what is best in this world and the world to come.
Allah Almighty knows best.