If I was to remove C1 and C2, would this schematic be properly filtered? I have heard a lot of conflicting information about this, but apparently the bluetooth module I'm using generated a considerable amount of noise that needs to be filtered out?
It looks all right to me, though I don't know which module you're using, so I don't know if that requires any kind of additional filtering. The datasheet is your friend in these cases
Also, should I decrease the voltage of the regulator I have in the circuit? The bluetooth module datasheet says that the operating voltage is 1.8. - 3.6, so what is the highest regulator rating I could have to get a consistant voltage into the circuit? Again, thank you so so so much for your answer!
Max
Well, it depends (it always depends!). The question is: what happens when the battery's voltage drops below the minimum required by the LDO to regulate its output? I think the most likely outcome here is that you'll get lower-than-expected voltage on the output, but, given that your device can work all the way down to 2V, you'll probably be okay.
However, you do need to worry about depleting the battery down to the point where it becomes damaged. If your battery already has a protection circuit, then it will automatically cut off before that happens, but if it doesn't you have to either be careful and never let it deplete to that point, or add some protection circuitry of your own.
—CC