I know that when you connect 2 identical batteries in parallel you get the same voltage but twice the capacity. So, my questions are 1. What happens if you connect 2 batteries that are not the same voltage? And 2. Should one use a diode when connecting batteries in parallel?
If your batteries are not at the same voltage, then the higher voltage battery will discharge into the lower voltage battery.
If they are rechargeable batteries, then this will balance itself out, just so long as the difference is not so great that one damages the other. It's always a good idea to make sure they are at the same level prior to connecting them.
A diode will prevent one from feeding into the other, but has a draw-back. It will not allow the batteries to be charged in-circuit, and they cause a voltage drop... which could result in up to 0.7V to be lost, and if the battery voltage is low to begin with, this will cripple your power supply. (ie... two AAs in parallel, with a diode on each will only give you about 0.8-0.9V even though the batteries are 1.5V)
And, when connecting them in parallel, the 'Current' is additive, the voltage stays the same, so you would get the sum of both batteries mAh, but you would also have a new 'battery' that is capable to putting out more Amps at a time. This is why some Diesel trucks use 2 batteries in parallel, so they can push well over 100A into the starter to get the engine moving.
Oh... if you do plan to use two different batteries, that have a different voltage, then a diode will, like stated above protect one from back-feeding into the other. This will essentially allow the lower voltage battery to be a 'back-up' for when the higher voltage battery finally dies enough that it's voltage drops down.