Must this switch be turned on and off periodically?
1) Yes: however, your driver must be isolated as the gate nodes need to be Vgs(on) above the common source node.  Photovoltaic isolator ICs or driver-transformer-rectifier combinations are usually used for this task -- either way, it's a switchable, isolated, low-current 10V gate supply.  You'll want to work out how much current the isolated gate supply must pull, though (Qg is your friend here), so that you can determine if a PVI IC will suffice, or if the more-parts approach (transformer driver IC, small transformer, and rectifier) is needed.

2) N/A -- 1) is certainly possible

3) A mechanical relay could be used instead; however, you would need to be very careful to snub the relay's contacts correctly lest you get a pile of arcs and sparks every time the relay switches.