It really does depend. A few tenths of a volt can mean a lot to an LED. One, because it's brightness is not a linear function of voltage. With an incandescent bulb, if you give it half of the voltage, it will typically go down half in terms of light output. That isn't the case for LEDs. Second, they can be very picky about voltage and most of the time overvoltage is what kills an LED prematurely. But yeah, 13 V is most likely going to kill the LEDs.
		
		
	 
Yes, and the forward voltages can vary widely.
Green and blue dies tend to use similar construction, and also have similar forward voltage ranges. (White LEDs use a blue die, with a phosphor layer over it.) Those are generally 2.5-3.5V, though I've seen some go all the way up to 3.99V. 
Then there are the red and amber dies, which usually share similar properties. They tend to have a lower Vf, but they too have a wide range. I've seen them vary from 1.80V up to 3.5V, with 2.00V being quite common, at least in my experience.
But yes, they are quite sensitive to the voltage. A small increase in voltage can drive a lot more current through the LED. 
Example, on pg16 (PDF). Note the scaling on the X-axis, particularly on the top graph. That only covers 1V, but you can go from 100mA up to over 1000mA in a big hurry. And to make things even more interesting, you've got the thermal runaway problem: As the die heats up, the forward voltage drops, meaning that the same amount of voltage across the LED will now end up pushing even more current through the die. So the die heats up more, and the forward voltage droops even more. You can quickly end up with a roasted LED.
I suggested 
Supertex's CL2 to Oldsmoboat awhile back for an application he had.
Connection diagram example.
It's a constant current regulator.
It acts as a smart resistor, dynamically changing its own voltage drop to ensure that 20mA gets through to the load. It does this by dumping out the unused power as heat, so it's going to get warm. In this case, the more LEDs in series you put on the end of it, the cooler it will be, because the voltage will get dropped across the LEDs, leaving less for the CL2 to dissipate as heat.
They've also got 
the CL220N5 now. Just to try it, I ran a CL220 + bridge rectifier straight off of 120VAC to power a single LED, though that's not necessarily efficient or safe. 
