First, your link does not work, so I have no idea of what power supply you are using. Simply recreating your link when on the web site produces no result. I'm assuming you were looking at their 30V/3A power supply PS300U3. This supply has no PWM setting, and if you applied 30 volts to your LED for more than 10 usec, yes you killed it. As for applying 15 volts, I suspect that you had the current limit set to 40 mA. At this point your LED was dissipating .6 watts, and if you did that for long you would have killed that LED, too.
Looking at the current curve, a quick approximation for voltage rise is to note that from 15 mA to 50 mA, the nominal curve rises 0.1 volts. 1.15 / .1 is 11.5 volts, so a rough estimate suggests 12 volts at 1.2 amps. Note that this is a peak power of 14.4 watts, and with a 1% duty cycle the average power is 144 mW, which is reasonable, since 1.6 volts time .05 amps is 80 mW - the two are within a factor of 2.
(1) Are LEDs able to take massive amounts of voltage when pulsed at
such short times, as long as the peak current stays below the limit?
Yes, indeed. Of course, you MUST keep the duration less than 10 usec, and the PWM frequency less than 1 kHz. Also, long term reliability may be bad. The data sheet just says keep the current below 50 mA, and if you want to do something else (like high-current pulses) you are free to do so. Just don't go crying to the manufacturer if the LED doesn't last long.
(2) As these voltage and current figures do not match the datasheet of
the LED, perhaps the lab source doesn't display the correct current
flow. How can we accurately measure this?
This is pretty straightforward. You make a setup like

simulate this circuit – Schematic created using CircuitLab
and monitor the voltages with an oscilloscope. A multimeter will not work.
You vary R1 while monitoring the scope V1 (1 volt equals 1 amp), and when you get a current you like, you can read the voltage across the LED (V2 minus V3). And whatever you do, don't use a pot for R1 - a 1 amp current will very likely burn the wiper. Turn power off, replace R1 with a different value, then turn power on again. Start with 50 ohms. Use 10 volts on the FET gate, and don't play with it. Make sure that the gate drive never stays high for more than 10 usec.
(3) What's the longevity of the LED when you operate it at the limit?
Does it have enough time in between pulses to dissipate the heat that
is generated when running at 1%?
Absolutely no way to tell other than by doing it. Probably not great.
(4) Is it possible to get a peak current of 18A @ 1% duty cycle out of
a 3A source without blowing it up?
With a good, current-limited supply? No. It won't blow up, mind you. It just won't provide more than 3 amps. With a cheap, voltage-only supply and a narrow pulse width? Sure, especially if you put a big capacitor on the output. Of course, this requires that you are not trying to provide the pulses by commanding the power supply.
With all of this said, you are going about this the wrong way. You need to stop and think about what you are doing. At the very best, your average current per LED will be 1.2 amps x 1% (your duty cycle) or 12 mA. And I can guarantee that the efficiency of the LED will drop at higher current levels, so you will get even less than this in terms of brightness. An LED is not a light bulb, where the light power is roughly the electrical power in. You will get more brightness by driving each LED to a maximum of 40 mA. Not 50 mA. 50 is the manufacturer's absolute maximum, and driving any component to its rated maximum is a good way to get reduced reliability.
EDIT -
1) Power Supply - The problem with the link is that Velleman apparently does not sell that model in the US, so it is necessary to select a European country in order to see it. However, this doesn't matter, it's just a switching supply.
You have misunderstood the current limiting circuitry, though. You might do well to contact Velleman and ask for their specification on response time to a current limit event. It is probably in the range of 50 to 100 usec. Not only that, but the high ripple voltage (200 mV) suggests that they don't do anything special on their output. It is just an inductor/capacitor combination. This means that when you pulsed your LED, the output capacitor discharged immediately into your LED, and the supply also provided a pretty good slug of current as well, while the current limit function never really engaged.
You need to follow mkeith's advice, and use a current limiting resistor in series with the LED.
2) Pulse Width - Your description of what you need is still unclear. As best I can understand it, you have an autonomous camera which takes 3 fps pictures, and are trying to provide IR illumination. At this point, you do not know exactly when each picture is taken or the shutter speed of the camera.
If this is true, PWMing the LEDs is simply not appropriate. Yes, by running the LEDs continuously you will waste power by illuminating the target area when the camera is not utilizing the illumination. However, since you don't know when that is, there is no sense worrying about it. Just run the LEDs at 40 mA and be done with it. Consider the situation where the camera takes 3 fps with a shutter speed of 1/100. If the LEDs are simply run continuously, each exposure will use only .01/.33, or 3% of the available light. If the LED is being PWM'd at 1 kHz, a single exposure will only use 10 pulses worth of light out of 333 which occur during 1/3 of a second. Efficiency is 10/333, or about 3%.
On the other hand, let's say you can either provide the shutter drive, or look at the camera data to determine when the camera has finished acquiring an image. This still does not tell you what the shutter speed is, so you cannot tell how short a pulse you need.
Note that the pulse condition (10 usec @ 1% duty cycle) says that as long as the shutter speed is greater than 1 msec, continuous illumination is the way to go. Like I said earlier, 1% of 1.2 amps is 12 mA, and 40 mA average for continuous is more than 3 times better, regardless of efficiency drops. The only exception to this is if you need shorter exposure times. If the camera shutter speed is less than about 300 usec, than pulsing the LED can be considered. And it's also possible to consider using very short LED pulses as a strobe light to freeze high-speed motion.
3) Efficiency - Efficiency is measured in optical output vs current, and all LEDs show a peak efficiency at (typically) a few mA. An article on the subject: http://www.electronicsweekly.com/news/components/led-lighting/provred-why-led-efficiency-drops-at-high-current-2013-08/. And here http://www.tech-led.com/data/L940-66-60-550.pdf is the spec sheet on a high-current illuminator. Note that the efficiency (mW/mA) is .875 at 700 mA, .800 at 5 A.
4) Voltage Drop - While your specific LED does not have a high-current spec for Vf, http://www.adafruit.com/datasheets/IR333_A_datasheet.pdf is probably a pretty good guide. The material (GaAlAs) is the same.
100% PWM duty cycle means the PWM pulse is high 100% of the time. This is the same as the flat level of whatever the pulse high voltage is. For a typical microcontroller PWM output, this would be its supply voltage.
In many cases, the average voltage of PWM is what matters. The pulses are deliberately fast enough so that their effect is averaged out. This is the case, for example, when driving LEDs, solenoids, and motors.
However, when driving a transformer, only some function of the AC component makes it to the secondary. DC just causes heating without any benefit, and brings the AC peaks of one direction closer to saturating the core.
Therefore, when driving a transformer with PWM, the maximum result is obtained from 50% duty cycle. 100% yields the same result as 0%, except for the extra heating. Of course this is taken into account when driving a transformer. The firmware, or whatever is adjusting the duty cycle, must be limited to not exceed 50%.
You might consider this the maximum drive level, but you'd still call it 50% PWM duty cycle.
Best Answer
Let me open my trusty MIL-HDBK-217F and see what it says about LEDs and their longevity: -
The main factor affecting the failure rate per million hours is temperature.
Of interest, if I read the next section about laser diodes they do take into account duty cycle pulsing but their conclusion (on page 6-21) is that at 50:50 duty cycle the failure rate for laser diodes is about 25% of that when continually driven.
They also conclude (on page 6-22) that if you operate a laser diode at a light output power of 50% of its rating it will last ten times longer than operating it at 95% of it's rated output power.