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.
It is an obligation to wash seven times, the first of which with soil, any container, clothes or anything else, that is licked by a dog, or touched with its urine or excrement, as the Prophet , said: "If a dog licks a receptacle of yours, one should wash it (that receptacle) seven times, the first of which with soil." This applies to a receptacle or anything else as the impurity is the same - which is the saliva of the dog - and its urine and excrement are worse than its saliva.
As regards how to purify a cloth which is licked by a dog, then if the cloth will not be damaged by the soil, then the first wash should be with soil mixed with some water so that the soil will reach the entire place touched by the impurity, and one should rub it and then wash it with water the second, third… until the seventh time. It is not sufficient for the first wash to simply put some soil on the cloth and then remove it with water (one must rub it). However, if the cloth could be damaged with the use of soil, then one may put a just a little bit of soil on it. Many jurists stated that soap and other cleaning substances take the place of soil. This is a summary of what the jurists mentioned about the method of removing the impurity of a dog.
With regard to saying Aameen (Amen) for a supplication made by a Christian mother, then some jurists forbade saying Aameen for a supplication made by a non-Muslim, but some others permitted it on the condition that there is no transgression in the supplication; this is the view which we adopt in IslamWeb. However, some jurists stated that it is desirable to say Aameen for a supplication of a non-Muslim if he supplicates for guidance for himself and for victory for the Muslims, for instance.
Therefore, it is permissible for you to say Aameen when your mother supplicates for good for you, and we advise you to invite her to Islam as Allaah might save her from the Hellfire thanks to you. Your mother is the first person whom you should endeavour to save from the Hellfire, and therefore you should call her to Islam with wisdom and soft words using any means that could be suitable for you as she is your mother.
Finally, we ask Allaah to open her heart to Islam.
Allaah Knows best.