Powering Arduino Pro Mini with different voltage, fuse setting
Read the datasheet once more. A certain voltage is needed for a certain clock frequency. There is a graph somewhere.

Since the Atmega328p is able to run up to 5.5V for VCC, that 5.5V is for every clock frequency.

Using 2V for VCC is not enough to run it at 16MHz. According to the datasheet even 3.3V is not enough. But 3.3V and 16MHz will work (it is outside the specifications though).

The RAW pin is the voltage before the voltage regulator, right ? Depending on the voltage regulator you can apply a large range of voltage.
See the pinout.

When you remove the voltage regulator, the RAW pin is open. Sometimes I remove the voltage regulator and use 3 AA batteries and feed the 4.5V into the VCC (or 5V pin) of a 8MHz ATmega328p board.


Yes. The bootloader and sketch have to know the clock frequency. A sketch compiled for 8MHz is for a 8MHz Arduino board and a sketch compiled for 16MHz is for a 16MHz Arduino board.
2V is too low. 3.3V should work.
2V is too low. 3.3V depends on the voltage regulator.
No problem. The voltage regulator gets a reverse voltage.
No problem.
No. See the schematic.
To VCC. 
Not a "must", but it is highly preferred to remove the voltage regulator, so no current is leaking towards the voltage regulator which gets a reverse voltage.
Yes. The fuses are to lower/disable the BOD, so a higher voltage is no problem.


This is the best tutorial about the power of the Atmega328p : Nick Gammon about Power

ADDED

I forgot to mention to remove the led as well.
This is a Pro Mini with a coin cell.

Lowering the clock speed will make it run at lower voltages, but that will not always reduce the used energy. Doing a calculation at a slow speed takes more time than the same calculation at high speed (and going into sleep mode after that).

ADDED 2

When a low voltage is supplied (for example 2V to a 5V 16MHz Arduino board), the voltage regulator will not work. How much current/voltage is still passed on depends on the voltage regulator. With the BOD disabled, and the microcontroller runs below its required voltage, anything weird can happen. Parts of the chip will work and parts will not. That is why the BOD was invented in the first place (to reset the chip and avoid weird things to happen). I think it is very unlikely that the microcontroller will get damaged with a (too) low voltage.