Can autothrottle be used without autopilot (and vice versa)?
They can be used individually, yes. When flight directors (or autopilot) are on then they dictate what auto throttle should do. But it's not a good idea to fly manually, use auto throttle and have flight directors still on and here is why:


When autopilot is off but the flight directors are on and auto throttle is on you can select a vertical mode like flight level change or vnav and then auto throttle will take commands from that selected vertical mode. In flight level change, for example, thrust is either full climb thrust or idle thrust, which means: If you as a pilot flying don't pitch for speed then you just have throttle stuck in either full or idle and you either overspeed or stall. Not a good idea to leave flight directors on and use autothrottle on its own!
When autopilot is on but auto throttle is off then autopilot and you select a vertical mode like vnav or flight level change then you are in command of delivering thrust, so you need to advance throttle to start climbing or you need to retard throttle to descent. And if you use vertical speed mode you are in control of airspeed. So you have to monitor your climbs and descends and airspeed but it's not too bad. In the end there are other aircraft types that have an autopilot but no auto throttle.
When autopilot is not engaged and flight directors are off then auto throttle controls just speed, this is safe. Airbus' landing checklist has an item for this: "auto thrust? - SPEED".


This applies to 777, 747, 737, A320, A380 and probably (have not checked the fcoms) A350, A330, A340, 757, 767