Hi,
I am having trouble figuring out this circuit. I know what the tank circuit is doing, but I get stuck at the FET with the capacitor across the source and drain. Could someone please help me to understand what is going on the with FET in this circuit and what the capacitor is across the drain and source is for? Please

 

Q1 is an oscillator/modulator. The capacitor provides positive feedback to sustain oscillations.

 

Can u tell me if the FET in the schematic below is doing the same thing?

 

Could be. Looks like it might be a regenerative receiver.

 

What is the source coil designed to do?

 

What is a source coil? Which circuit are you asking about? I don't know much about the second circuit, so I'm not going to comment further on it. Maybe someone else can help.