Max voltage is 5.5v per datasheet.
Could probably drop 0.7v through a normal (non-schottky) diode to get 4 AA's into safe territory. I would not add a schottky diode after that; 5.3 is safe, and that'll give you more headroom as the batteries run down.
Bypassing the regulator (by supplying 5v directly) will not reduce the current (though since you don't need to give it an extra 2v for the dropout, the power (ie, volts times amps) is lower)
Removing the regulator from the board entirely will slightly reduce power consumption, but the quiescent current of the regulator isn't that high - it's only relevant when the chip is sleeping and you've already removed the power LED.
Spec for 16mhz is ya, a little under 4v. In practice, under normal conditions, you can generally run a 16mhz '328p down to 3.6v no problem (which is relevant as that's the min output voltage for a LiPo battery)