That is not a coil, it is a transformer. Unless you are trying to resonate with the large magnetizing inductance of the transformer, there is no LC circuit, only a C circuit with a transformer.
Yes, with the correct approach. However, as has already been mentioned, not with the method you proposed.
An unbalanced passive frequency doubler or tripler can be built with acceptable conversion loss to avoid the necessity of baluns. The basic blocks of the circuit are an impedance matching network (LC), the diodes and an appropriate LC bandpass filter matched to the desired output impedance.