"Mother lives at home."
Translation:माँ घर में रहती हैं।
Is it possible to use रहते हैं instead of रहती है in this sentence, as to express it in a polite manner?
No. It has to be रहती हैं since रहती can be used for both singular and plural feminine words. But it is true that हैं has to be used to indicate that it is plural (or in this case singular but polite). रहते would be used if you for example said father instead of mother, as it only applies for masculine words.
Not really. "घर में" is a perfectly valid expression and in the given sentence, it's as appropriate as "घर पे/पर", if not more.
I have mostly seen it in sentences where it means "in" For example पक्षी घास में हैं = the birds are IN the grass.
I suppose it can also mean "at", mother lives AT home