Could a routers WiFi signal be amplified and used from a distance? Something like a radio stations output wattage.?

You can use "repeaters" or "range extenders" to pick up the signals, and re-transmit them. That works reasonably well, at the expense of latency.

You can't legally boost the transmit power from the router. (And it wouldn't do you any good unless you boosted the transmit power from your laptop / phone, as well.)