It's hard to say where your bottleneck is regarding your streaming issues, but the 1000Mbps should provide a noticeable improvement in general surfing/streaming. I suspect that the 1000Mbps service is fiber so pings should be lower, allowing for a "snappier" feeling when surfing. 1000Mbps should provide more than enough bandwidth for your entire family's streaming needs.
If the new service is indeed fiber, new wiring will need to be run from the pole to the ONT, then to the router. You can then either use cat5/6 to connect your pc to the router or simply use wifi.
Home phone wiring is not compatible with fiber so that will no longer be used.
For +$20/mo, I'd upgrade in a heartbeat.