How does this circuit multiplex an antenna between a transmitter and receiver?
In your excerpted schematic, you left out the rest of the receiver input, which includes a DC path to ground through the coil L4 and the primary of the transformer T5. That's why you can assume the source of Q10 is at DC ground.

Depending on its value, C27 is probably there to provide an AC impedance that helps to isolate the switching circuitry from the low-pass filter in the transmit path (which is there to reduce harmonic emissions). It's a compromise between providing sufficient isolation while transmitting and providing sufficient signal when receiving.

Yes, you do need to pay attention to the impedance issues (see answer about C27 above). One of the things that C27 does is give you a more definite value of capacitance to incorporate into the design of the transmit path.

The main factors for transistor selection would be the peak voltage that it (Q10) can withstand when off, and the terminal-to-terminal capacitance, which, if too high, will limit the amount of isolation you can achieve. Note that you can't use a "power MOSFET" in this application; its body diode will conduct on the negative peaks and create problems.