1. On a fixed-gear bike, if the wheel that's being driven by the chain is turning, then the pedals are turning. True in either direction. The cog is "fixed" to the hub. If the hub is turning, then so is the cog, the chain, the pedals, ... you get the idea.
2. A singlespeed is a bike with only one gear (i.e., one chain ring and one cog).
3. Here's where it gets tricky: Most fixed-gear bikes are singlespeeds. But not all of them. You can have dingle cogs and various other such contraptions that let you run a fixed hub with multiple gears on it. And if your hub is threaded on the non-drive side, you can put a different cog on there and then flip the wheel around to run the second gear.
4. Even more trickiness: Singlespeeds, in the way in which that term is used in common parlance in the United States, are not fixed-gear bicycles. Calling something a singlespeed usually means that it has only one gear AND a freewheel hub. Similarly, if you hear "fixed gear," what most people (again, at least in the U.S.) mean is that the bike has only one gear and that gear is fixed to the hub.
5. What's it all mean? My $0.02 is that riding a SS distance is considerably easier than riding a FG. However, ascending a hill -- especially a very steep one -- is actually easier on a FG. On a FG, the bike will help put the next pedal in position because it's rolling every so slightly forward. You get a slight bit of rest there and, believe me, every little bit helps when climbing a whacky pitch when you're severely over-geared. On a single speed, you've got to put that pedal into position. You can rest, but the pedal will not move forward without you doing the work.