The 20mA rating of your LEDs is probably the recommended maximum current. Most LEDs will produce light at much lower currents - I find 5 mA is sufficient for indicator light use.
Try the LEDs with different resistors to see what current is actually required to produce sufficient light for your application (more current = brighter light).
Also, check the maximum current the Pi GPIO pins can handle - I have a feeling it is less than 20 mA, but can't check at the moment.
Your "LED Driver" is most likely bad
The driver is clearly out of spec and mostly likely internally damaged. From you photos it isn't clear how the rest of your lights and power sources are connected, but you may have made an error here (I've even seen people connect the AC power line to the DC output side of the supply).
You have a current source, not a voltage source
If you look at the label you will notice that current is specified precisely (600mA) and voltage is specified as a range (16v-28v). You will also notice that the drawings on the label show a single current loop and specify 7 3-Watt LED's in series.
That you provided this equation indicates that you are confused about the difference between the two types of sources:
$$R=\frac{V_{Driver}-V_{LED} \times 7}{I_{LED}} = \frac{35\ V-3.7\ V \times 7}{0.7\ A} = 13\ \Omega\ $$
In your equation, you cannot know the value of $$V_{Driver}$$ as it is determined by the network (it's somewhere between 16 and 28 volts if an acceptable load is attached). Only the current value is constant in the normal operating condition.
Some background
A voltage source presents a single output voltage, no matter what you connect to it. To make that a true statement it will output any current the load wants up until it is incapable of outputting any more (current-limit or failure). Most people are familiar with this behavior as it is intuitive and commonly encountered.
A current source will attempt to output a constant current no matter the load attached to it. It will do this by adjusting the output voltage until it either can raise it no further (limit of it's upward adjustment range) or lower it no further (it will produce insufficient voltage to operate itself and shutdown).
This works via Ohm's law (V = I R) such that increasing the voltage will increase the current flowing and decreasing the voltage will decrease the current. The system is active and senses it's output current (while adjusting its output voltage) until the output current equals the number printed on the label (in this case 600mA).
If nothing (or too little load) is attached, it will output it's maximum voltage as it keeps trying to increase voltage to get increased current... and vice versa if too much load is attached.
Driving LED's
If your LED's are connected as parallel strings of series lights, they will need to be driven by a voltage source. This configuration is cheaper to design and build, but more difficult to install and subject to greater line losses as the LED strings get bigger (since you need to bring the full voltage of the power supply all the way to the end of the line).
If your LED's are connected in series (one to the next), then the same current that drives one light will drive the next. This configuration is used in most higher-end architectural lighting. You use a current source to ensure that no matter how many lights are on the string, the current output remains the same. The advantages are that you can easily add lights to existing strings without worry. The fact that current flows through all lights ensures that voltage losses in the line are minimal (most efficient power distribution). And, LED light output is proportional to current so a current driven approach best ensures uniformity of light output.
There are some limitations however. Current drivers are more expensive as they are produced in lower quantities and have to be matched to the exact fixtures being used. The fixtures must all be the same so that they have the same light-to-current relationship and will not be too dim (or burn up) under the constant current value applied. Series wiring of light fixtures is inconvenient in some installations.
Best Answer
Figure 1. Typical IV curves for various colours of LEDs. Image source: LEDnique.
Replacement of any of those small indicator LEDs with 3 or 5 mm LEDs should be fine.
A little bit of background theory may help. LEDs have a non-linear relationship between applied voltage and current. The forward voltage also depends on the colour as shown in Figure 1.
Your measured 3 V LED voltage suggests that you've got either white or blue LEDs on the board. With the 1 kΩ resistor in series the current will be limited to a safe value even if you change to one with a lower forward voltage.