Is this transistor switch circuit designed correctly?
1) The 1k ohm resistor IS necessary. The base - emitter junction of a transistor is a diode, and the voltage drop of a silicon diode is ~0.7 V. The transistor will allow as much current to pass as it need to to clamp that voltage to less than 0.7 V. This will interfere with the signal, as the RGB signal is indicated to the receiving device as a signal greater than 1V (1-3 V according to wikipedia: https://en.wikipedia.org/wiki/SCART).
In order to keep interference to a minimum, I would probably use an analog buffer (https://en.wikipedia.org/wiki/Buffer_amplifier#Voltage_buffer_examples) instead of a transistor to turn on the LED, but if you find it doesn't interfere, there isn't much reason to change it. 

2) If you find that pin 21 works as a ground for you, it shouldn't cause any damage. Looking at the pinout on that same wikipedia page, I would go with pin 18 instead. Pin 18 is specifically a ground for the signal on pin 16. Either way, a common ground is necessary for the signal to be able to turn on the transistor.