Flushing out a length of cable housing well is tricky. I don't think I'd bother trying. I'd think you'd need to pull the cable, and use a syringe or something to get any useful flow through it.
Brake housing has to stand a fair bit of compression forces, so the longer housing run you use the mushier the brake becomes. It is possible though to run the cable enclosed pretty much all the way. The big name back in the day was Gore Ride-On or something like that. You still use all the cable stops etc, but you use special ferrules which connects to a thinner sleeve that bridges the open bit between the cable stops.
It is often debated whether modern cables benefit from being lubed or not. Cables are usually stainless, and there is a really slick liner inside the housing. Not like before, when there was a galvanized wire running inside a bare steel spiral coil. Some still do lube, other claim it's just an open invitation to grit etc.
Only place I find factory lube is at the brake noodles.