Electrical – Passing Voltage above threshold

If by "best" you mean "most efficient", then you want a switched-mode power supply, or more specifically, a buck converter. Efficiencies above 70% are common, and above 90% is possible with careful design and selection.

With such a device, you may indeed be able to use a 12V battery without any more loss than you would have with a 6V battery.

The LM317 is a pass transistor driven by an error amplifier, essentially an op-amp plus a transistor. An op-amp plus a MOSFET is not very different, and you will not improve on the efficiency of an LM317 with any combination of op-amps and MOSFETS that don't involve rapid switching, and probably an inductor. Without switching, you have a linear regulator, and the only place excess voltage can go is into heat.

Electrical power is the product of current and voltage:

$$ P = IE $$

and as long as you have any device with a voltage across it and current flowing through it, you are converting electrical energy into something else, usually heat. Non-linear regulators avoid this problem by rapidly switching between periods of high voltage and low current, and low voltage and high current, except in an energy storage device (an inductor) which can store the electrical energy in something more easily recovered than heat (a magnetic field).

The threshold voltage can be increased if the source is not connected to the body terminal. The threshold voltage is

$$V_T = V_{T0} + \gamma\sqrt{2\phi + V_{SB}} - \gamma\sqrt{2\phi}$$

where \$V_{T0}\$ is the threshold voltage when the source-to-body \$V_{SB} = 0\$, and \$\gamma\$ and \$\phi\$ are device parameters. If the NMOS source is connected to ground and so is the body then \$V_{SB} = 0\$ and \$V_T\$ is minimized (it's a similar argument for the PMOS).

So, yes, it is possible to adjust the threshold voltage by not connecting the NMOS body to the negative supply and the PMOS body to the positive supply.

However, this is usually not done intentionally. You typically want to minimize \$V_T\$ -- for example, this would allow you to use lower supply voltages.

The body effect is particularly undesirable for a common drain amplifier because it lowers the voltage gain. Without the body effect the unloaded voltage gain of a common drain amplifier is

$$\frac{v_o}{v_i} = \frac{g_m}{g_m + \frac{1}{r_o||r_{oc}}} \approx 1$$

where the approximation assumes the resistances are large. However, with the body effect the unloaded voltage gain is reduced:

$$\frac{v_o}{v_i} = \frac{g_m}{g_m + g_{mb} + \frac{1}{r_o||r_{oc}}} \approx \frac{g_m}{g_m + g_{mb}} < 1$$