In the picture, the board is powered by USB, which allows for more current then the 5V regulator can supply.
Also, as mentioned on the product pages for LEDs, an LED draws approximately 50mA when red, green, and blue are set to full brightness, which would result in a bright white light. This is not how the LEDs are being used in the picture.
In general, the longer lead on an LED is the positive lead; however, since you cannot determine which lead was longer with them already cut, I recommend using the continuity test on a multimeter to determine which lead is the anode.
The LED voltage is usually determined by the color it emits, so you might try searching for an LED color voltage chart to determine its rated voltage. For the current, you will probably have to look it up in its datasheet.
It is possible to destroy an LED by connecting the polarity backwards, but the voltage required to break it is usually significantly higher than the forward voltage of the LED.
Using a variable resistor to determine a safe resistor value might work; however, it is probably easier to just buy new LEDs so you can get all the necessary specifications and calculate the appropriate resistor value.
I am sorry you are having problems with your VNH5019 shield. However, the blog is not an appropriate place for troubleshooting your problem. Instead, you might post this on our forum with more details about your setup.
The A* 32U4 should work with many of the Teensyduino libraries. The biggest obstacle we expect is that the pin mappings will be different. For others that are interested, we are discussing using the Teensyduino Keyboard library with the A* in this forum thread.
Feel free to keep us posted on any of your progress involving other Teensyduino libraries and the A-Star.