Last week a dog came to my house.

I usually take dogs who that are in an animal shelter to my house as a volunteer.
They are lost dogs. Most of them are very thin, dirty, don't believe trust in humans.
I take care of them. Sometimes it takes a long time to make them healthy and open their hearts.

Then I look for their new foster parents.

All the dogs I took have taken care have found their a new home house. I'm so happy when I know they will have a spend happy life with their new owner.

I live with my mother. She wanted to have her own dog. I hesitated because she is old. If she will be becomes sick, I have to have her dog until that dog's last moment of life (it would be better to say: I will have to take care of that dog forever).
I'm not sure I can live for over 10 years. ( Nobody knows though..)

Last week, I have decided that my mother has should have a dog.

A dog who that came to my house is a cute little mixed shiba..

Very well done, just a few grammatical errors here and there.

Your original text was, however, completely comprehensible before corrections were made.