Yes, the county sheriff issues them and they are valid state-wide.
Don’t know about the requirement to live in the county of application. But MN is a shall-issue state so there should be no need to apply in a different county to get a favorable result.
I could live and apply in county X and then move to County Y and be fine until renewal.
I was asking because it appears Castile grew up in Ramsey County, but had lived in Hennepin County at some point. The shooting also occurred in Ramsey County.
(An aside: I lived there long ago, but before the CCW/CHL was enacted. So, I know the cities and counties in the Twin Cities)