li- ion battery discharge query
The battery will not act as a current source, but if you connect it to a constant current load that draws 1A, the battery will in theory be able to deliver current for 2 hours. 

Chemical and resistive losses makes the real-world number lower than the theoretical one. And the more current you draw (the more amps), the higher the losses will generally be.

The voltage not coming into the equation makes Ah a less useful metric than Wh for many applications, but that's the convention. Especially when you're using the battery with constant power loads, you won't care as much about the current capacity as the power capacity.

If you know the nominal voltage of the battery, you can multiply the nominal voltage (V) by the current capacity (Ah) in order to get the power capacity (Wh). Using your example, 3.7 V * 2 Ah = 7.4 Wh.