If I run a Cat6 ethernet cable from my Linksys router to an unmanaged Netgear GS605 switch located in a building away from the dwelling housing the router, will I get wifi in the building?
No, you will not. The location that has the Lynsys Router will have the WIFI services. This is because the device that you named as the un-managed one, does not provide any WIFI services according to the specifications.