1. My grandma lives in the village.
2. My grandma lives in the countryside.
I don't think either of these would be used in the US.
In the US, people would say, "My grandma lives in the country."
"My grandma lives in the Village" presumes that there is a place that we have been calling "the Village," (such as the section of Boston called "Adams Village" -- not a real village at all, and not in the countryside), and I want to inform you that my grandma lives there. There is no "generic" place called "THE village" like there is a generic place called "the country" or "the countryside."
If I wanted to say "My grandma lives in A village," I think in the US people would say, "My grandma lives in a very small town."