A new Muslim whose mother has a dog and she is asking about a number of rulings that have to do with tahaarah (purification)
Question

Im a Muslim convert Alhamdulillah. My mother and my stepfather have a dog. My husband and I never eat anything they cook, neither do we drink water from their glasses. Also, if my mother wants to give us something we always ask her if she touched that thing with her wet hands, because we are afraid that the impurity may be transmited from her hands to that object.
Is it ok how we are acting or are we exaggerating? Are we allowed to eat from what she and her husband cook and use their vessels? Are we allowed to use their bathroom and wash our hands there and use their towels to get dry? Am I allowed to use a cloth that she wore before and she washed it in the washing machine or should I first clean that cloth using mud?
Also, if my mother and her husband come to visit us can they wash their hands in our bathroom, use our towels and eat and drink from our vessels, or would all these get contaminated by the impure saliva of their dog, after they touch them with their wet hands?
Praise be to Allah.Firstly: 
The najaasah (impurity) of the dog means that it is impure in and of itself. This has been discussed previously in the answers to questions no. 13356 and 69840. 
If a person touches a dog and his hand is wet or the dog is wet, then his hand has become impure and he has to purify whatever has been touched by it of garments, vessels and so on. For information on how to purify it from the impurity of the dog, see the answer to questions no. 41090 and 46314. 
Secondly: 
It is not appropriate to ask a kaafir about the purification of his hand or garments or vessels, because the basic principle is that they are pure unless it is proven that they have become impure. However it is better to avoid using their vessels and garments until they have been washed. Al-Hajaawi (may Allah have mercy on him) said in Zaad al-Mustaqni‘: It is permissible to use the vessels of the kuffaar, even if their meat is not halaal, and to use their garments if it is not known what state they are in. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: If someone were to ask, what is the evidence for that? We would say that it is the general meaning of the verse in which Allah, may He be exalted, says (interpretation of the meaning): “He it is Who created for you all that is on earth” [al-Baqarah 2:29]. Moreover, if Allah has permitted us to eat the food of the People of the Book, then it is well known that they sometimes bring it to us cooked in their vessels. And it is proven that the Prophet (blessings and peace of Allah be upon him) was invited by a Jewish boy to eat some barley bread and fat, and he ate some of it. Similarly, he ate some of the poisoned mutton that was given to him (blessings and peace of Allah be upon him) in Khaybar. And it is proven that he (blessings and peace of Allah be upon him) and his Companions did wudoo’ from the waterskin of a mushrik woman. All of that indicates that whatever has been handled by the kuffaar is taahir (pure). 
With regard to the hadeeth of Abu Tha‘labah al-Khushani, according to which the Messenger (blessings and peace of Allah be upon him) said: “Do not eat from them (vessels of the kuffaar) unless you cannot find anything else, (in which case) wash them and eat from them”, this indicates that it is better to avoid using them, but many of the scholars interpreted this hadeeth as referring to people who are known to handle and deal with impurities such as pork and the like. They said: the Prophet (blessings and peace of Allah be upon him) forbade eating from their vessels unless we cannot find anything else, in which case we should wash them and may then eat from them. This interpretation is good and it is in accordance with the guidelines of sharee‘ah.  
The words “and their garments” mean their garments are permissible. That includes what they manufacturer and what they wear. Their garments that they manufacture are permissible and we do not say that perhaps they were woven on an impure loom or dyed with an impure dye, because the basic principle is that things are permissible and pure. Similarly, clothes that they have worn are also permissible for us to wear, but in the case of those whom it is known do not take care to avoid impurities, such as the Christians, it is better to avoid wearing them, based on the implications of the hadeeth of Abu Tha‘labah al-Khushani (may Allah be pleased with him). 
The words “if the condition is not known” may be understood in two ways: 
1.That they are known to be pure
2.or that they are known to be impure.
If it is known that they are impure then they should not be used until they have been washed. If it is known that they are pure then there is no problem. Rather the problem arises when their condition is not known. Should we say that the basic principle is that they do not take precautions to avoid impurities and they are haraam, or should we say that the basic principle is that things are pure unless it is proven that they are impure? The correct view is the latter. 
End quote from ash-Sharh al-Mumti‘, 1/82 
An-Nawawi (may Allah have mercy on him) said, after mentioning the ruling on using the garments and vessels of the kuffaar: What we have mentioned of the ruling that the vessels and garments of the kuffaar are pure is our view and the view of the majority of the early generation (salaf). Our companions narrated from Ahmad and Ishaaq that they are impure because of the words of Allah, may He be exalted (interpretation of the meaning): “Verily, the Mushrikoon (polytheists, pagans, idolaters, disbelievers in the Oneness of Allah, and in the Message of Muhammad ) are Najasun (impure)” [at-Tawbah 9:28]. And because of the hadeeth of Abu Tha‘labah and the words of the Prophet (blessings and peace of Allah be upon him): “Wash them.” Our companions quoted as evidence the verse in which Allah, may He be exalted, says (interpretation of the meaning): “The food (slaughtered cattle, eatable animals, etc.) of the people of the Scripture (Jews and Christians) is lawful to you” [al-Maa’idah 5:5]. And it is well known that they cook their food in their pots and touch it with their hands. And because of the hadeeth of ‘Imraan and the action of ‘Umar which are mentioned in the book. And because of the basic principle that things are pure. And because the Messenger of Allah (blessings and peace of Allah be upon him) used to give the kuffaar permission to enter the mosque; if they had been impure, he would not have given them permission. 
End quote from Sharh al-Muhadhdhab, 1/320 
To sum up: 
There is no reason why you should not use the vessels, towels and garments of the non-Muslims, although it is better to avoid doing so especially if they are known to handle impure things. If the cups and vessels that your mother uses are kept in a cupboard that the dogs cannot get to, there is no reason not to use them. The same applies to towels and garments; if your mother washes them in pure water, there is nothing wrong with using them. There is also nothing wrong with your mother washing her hands in your bathroom or eating and drinking from your vessels, because the basic principle is that her hands are pure. 
And Allah knows best.