Re: questions about mixer
Yes, is possible to design this kind of mixer, but is rarely implemented.
In this case is preferable to use a BJT instead of a FET. BJT due to its impedances is easier to be matched.
The most critical is the input matching circuit, that should not only match the RF source to BJT base-emitter junction, but must also provide an IF short circuit to the device.
If the IF frequency is much lower than the RF, this short can be realized via the bias-circuit elements. As long as the IF short is realized effectively, the only other critical function of the input matching network is impedance matching at the RF frequency and, if possible, at the LO frequency.