How can I clean hard water buildup from pipes?
I'm wondering if the problem isn't mineral buildup, but rather sediment in the lines. Not sure if this would cause a failure in your softener, or if it could be caused by the softener failing.

The sediment in the pipes would go to various fixtures and get stuck in the filters and aerators. You'd have to remove each one and clean it out. On sinks, this is usually part of the aerator and may unscrew without major tools. On shower heads, this may be where it screws into the elbow pipe. And on toilets, newer ones have a cap at the top of the supply pipe that you push down and give it a quarter turn (turn the water off first, with the cap off and a cup over the top to keep water from going everywhere, run some water to clean it out, and then reassemble).

For the hot water heater, you would need to simply flush it out (which should be done periodically due to natural buildup). Attach a hose to the drain on the bottom, either run it outside or into one of your drains, turn the heater off so you aren't trying to heat water that's being discarded, and open the drain for a bit. If you're observing the outflow, keep going until it runs clear. Otherwise, I'd guess 5 minutes of draining every year should be good, but that's a guess.