A:
The Chromecast dongle is a device on it's own, it has its own WiFi radio, independant of the WiFi adapter that the new TV may have. If the Chromecast has a better range than your TV, this might be misinterpreted as "the chromecast enhancing the WiFi signal" which is not the case.
To be a cord-cutter you need a reliable WiFi connection, because the Chromecast and the phone you'll be using have to connected to the same network simultaneously in order to work. Also, with a weak WiFi signal, I'm not very confident that you'll have a seamless experience streaming content to your TV.
If you have any questions, @JerryZaz on twitter.