Q:

How to determine which diode conducts in diode-OR application

Say I have two 12V power sources - 12V_A and 12V_B. What happens in the following scenarios:

If 12_A = 12_B = 12V. Do both diodes conduct?
If 12_A is slightly higher than 12V_B. Does the top diode conduct? Or both?
If 12_A = 12V and 12V_B = 0. We know that top diode conducts for sure, but is there also leakage through the bottom diode?

A:

You have to understand diodes are resistors that change value depending on the voltage across them.
You should be familiar with the voltage current relationship of a diode.

But if you take that graph and plot the slope versus the voltage, you get V/I which is the resistance vs voltage graph, you are likely less familiar with and looks something like this.

So, having that in mind, lets look at your cases.
In case 1, if they are matched diodes, they will both have the same voltage across them and will both conduct equally. In reality they will differ and whichever diode has the smaller threshold voltage will conduct a lot more.
In case 2, the top diode will have the higher voltage across it and will conduct more than the bottom one. What the bottom one does depends on your definition of slightly. If the right side of the diode is lower voltage than the left it will still conduct but at a higher resistance. If the voltage is reversed it will leak a bit.
In case 3, obviously the top diode is conducting, the bottom diode will either present a large resistance and leak some current back to the left, or the voltage will exceed the reverse breakdown voltage and it will conduct back to the left and pull down the output.

