You have to open every endpoint, or the crap in the line to any individual endpoint will be left undisturbed by the water passing by on its way to the "furthest endpoint." For effective flushing you might need to disconnect faucets to get sufficient flow-rate to move the crud. You can apparently check progress by looking at the pipes to see if you need to do more. You'll probably need to open each endpoint individually or the pressure & flow will be insufficient to make a dent.
Might as well install basic bulk filtration before bothering with this - I'm personally rather happy with the filter style (spin-down) that combines centrifugal force with a filter element, throwing the bulk of material out and down rather than just dumping it straight onto a filter element and plugging it. Versions with a collection area and valve below the filter element are available and simplify cleaning that bulk material out.
If you want finer filtration that those provide, place it after them, their life will be increased by the bulk removal provided, but for high-flow flushing you don't rally want a very fine filter in place as it will impede the flow.
Water heaters have a drain valve and a cold water inlet. You clean the water heater the same as ever by opening the drain valve (usually after attaching a hose to drain somewhere convenient) and the cold supply and flush until it runs clean. Be sure to get all the air out of the tank via the hot water pipes before you turn the heater on. Since the heater is the point where hot joins cold normally, there's really not a lot of point in "joining hot to cold, flushing pipes, then reconnecting the water heater" - just connect the water heater (rather odd for it to be disconnected and capped off - since it sounds like a new house you are moving into, that makes me wonder what the seller(?) was up to) and flush it, and flush through it for the hot-side pipes.
It might be worth doing some "blow-out" with compressed air interspersed with water flushing, depending how just flushing with water goes. Or combine water and air (bubbles help stuff to move.)
If veering off into adding chemical agents, stick to ones normally used in potable water systems.