Can you have a router and pick up wifi from about a half a mile down the road.
and what kind
Not under normal conditions ... *maybe* if you put a wifi signal extender on your roof, and have a line-of-site that far away ... but wifi frequency and power are designed for short range, not even 1/10th of a mile.
It is possible to pick up Wi-Fi from a radius of half a mile or more using a long distance Wi-Fi router. In addition, any brand of long range Wi-Fi router can easily cover the half mile.