Running a 2V motor from a 12V H bridge is perfectly fine as long as you don't exceed 1/6 duty cycle, at least not sustained.
No, you don't want the transistors to drop voltage to make up the difference. That is difficult to manage exactly, and would just burn up the extra voltage in heat. You want the transistors to be either fully off or fully on most of the time. That way they dissipate as little power as possible. Basically the duty cycle provides the "gearing" so that power in is roughly power out minus a little loss. This is in effect a switching power supply where the motor itself becomes one of the components. With a PWM frequency of a few 100 Hz or more, the motor only "sees" the average voltage and won't be hurt with 12V pulses for 1/6 of the time instead of steady 2V.
Let's say the motor draws 500 mA at 2V when running normally. That's 1 W. If you drop 10 V by having the transistors be partially on, the current draw from the 12V supply will still be 500 mA. However, that is 6 W. The motor only gets 1 W, so the remaining 5W will end up heating the transistors. Not only is does this waste 5/6 of the power (only 17% efficient), but 5 W heating the transistors is something you have to think about and handle. At the least that will require some heat sinking. Getting rid of waste heat is bulky and expensive. Electronic means to avoid making the heat in the first place will be smaller and cheaper.
Now consider how this works out with a controlled duty cycle H bridge. Let's say the two transistors that are on together drop 1/2 Volt total. That's 11.5 V apparent voltage that is being switched on the motor. The 11.5 Volts will be applied to the motor 2/11.5 = 17% of the time. The average current draw from the supply will only be about 90 mA instead of the 500 mA average that will go thru the motor. That 90 mA accross the 500 mV drop of the transistors will cause them to disspate only 45 mW. You'll barely notice them getting warm. In this example, the efficiency is 93%, which is doable although perhaps a bit optimistic for someone that hasn't done this before. Still, even at 85% effciency (quite doable) there is only 177 mW heating the transistors. Again, that is small and doesn't require any special mounting or cooling.
At this low voltage, it is probably easier to use logic level FETs for the low side since these can be driven directly from the digital PWM output of the micro or whatever. You didn't say what current the motor needs, but since it's a 2V motor I'm figuring only a few 100 mA in normal operation. The IRLML2502 is a nice little FET that should work well for the low side switches. The high side switches can be easily done with PNP power transistors since even a few 100 mV of drop won't be a problem. In that case I would leave the appropriate top switch on continuously when the motor is set to a particular direction, and PWM the low side switch. That will be easy to do and quite efficient. The right P channel FETs will work even better but you have to think more carefully about how they are driven.
Another advantage is that you can exceed the apparent average 2V on the motor for short times as needed. You don't say what your application is, but this could be useful at startup, to overcome a short time sudden load, etc. If this is servo controlled, then having the extra torque headroom will be useful even if it will only be used a small fraction of the time.
