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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, we praise Allaah Who guided you to Islam and we ask Him to make us and you firm and steadfast on it until the Day we meet Him.
Since the situation in your family is as you mentioned in the question, if you can find a separate accommodation then this is better, but at the same time you should be kind to them and endeavor to invite them to Islam in a wise manner and with good admonition, especially your mother.
However, if it is difficult for you to get a separate accommodation, then you should avoid as best as you can touching the items in the household that you are certain are impure, and Allaah does not task a soul beyond one's scope.
You should not deem your clothes or your body to have become impure based on suspicion alone, as they are presumptively pure. The fact that there is a dog in the house and other matters which you mentioned in the question are not enough to deem that everything in the house is impure.
If we presume that the household items are evidently impure due to the impurity of the dog, then the presumptive principle is that those items are pure because in case there is a conflict between the basic rule (the Asl) and the apparent condition (the Thaahir), the basic rule takes precedence.
An-Nawawi said: "The thing about which neither the impureness nor the pureness are certain, and the like of which is that it is generally impure, there are two pronouncements concerning it (in the Shafi'i School), because of the conflict basic rule of something and its apparent condition. The most authoritative of the two (pronouncements) is that it is pure pursuant to the basic rule."
Nonetheless, it is better to have some special dress for the prayer which you only wear when you pray, so that this dress can be far from the impurity of the dog and from what you suspect to be impure, in order to feel assured that you have prayed in a pure dress.
The prayer is not rendered invalid just because a dog walked on the place of the prayer, and the place does not become impure just because the dog passed through it, as dogs used to pass in the mosque at the time of the Prophet sallallaahu ‘alayhi wa sallam, and they (the Prophet sallallaahu ‘alayhi wa sallam, and the Companions) did not sprinkle water there, as Ibn Umar said; and this was reported by Al-Bukhari. The place is ruled to be impure only if a dog urinates on it.
Finally, you are not obliged to wear shoes at home, but if you do so, this is better for you.
For more benefit, please refer to Fataawa 92109 and 228397.
Allaah Knows best.