Isn't it murder going up hill on a fixie, and even worse when going down?
I find going uphill much easier and faster because you have to attack the hill instead of just dropping down through the gears. Long steep hills can be a struggle but I have yet to find one where my speed has dropped below 9/10mph and I'm running 75.7".
Going downhill has a learning curve but it doesn't take long before you can get you're legs spinning at crazy speeds or controlling the speed with you're legs.
Are there any advantages to them over free-wheel bike (other than you can cycle backwards)
Very low maintenance, increased low speed control, the momentum helps to pull you along, teaches you to spin at rates you never thought you could manage, works your muscles in a different way when braking.
What possible advantages can a single-speed bring?
I can't see any, other than less maintenance.
Why should they cost more than a decent geared bike?
They don't have to, you can build one for less than £100 by converting an old bike. Cheap new ones come at under £200. However, when buying new you get what you pay for, assuming you avoid the fashion bikes that sell for silly inflated prices due to the current 'fixie' craze.
I love the Fixed and ride it all the time, it just feels a much better ride being part of the bike, being in more control and not having to worry about maintenance regardless of the mileage I rack up. If something does need to be replaced I can replace it myself in minutes and be ready for the next ride. I've only been out on my MTB a couple of times since I've had the Fixed.