Maximum length for ethernet is 100 meters, so you will need at least two switches and maybe three for better reliability.
You can probably do what you want by using DMZ or port forwarding, but it would be simpler to keep everything on the same LAN by using the second and third routers as just ethernet switches and, optionally, wireless access points.
You can probably do what you want by using DMZ or port forwarding, but it would be simpler to keep everything on the same LAN by using the second and third routers as just ethernet switches and, optionally, wireless access points.