Ultra low power arduino wireless sensor : care of battery discharge
In regards to the questions asked, you need the capacitor C14 to provide a low impedance input to the ADC during sampling.  If you look at how an ADC module works, there's a capacitor that holds a charge while the ADC conversion is taking place.  Before that can happen, this capacitor needs to charge to the value supplied by the pin.  This is called the sample time.  In order to do this, current must flow into this capacitor.  This is where the external capacitor (C14 in your case) comes into play.  When C14 is present, the current, or charge, primarily transfers from C14 to the internal capacitor while the voltage across C14 remains stable.  When C14 is removed, the current must flow through R1, which is a high impedance of 10k, which causes a voltage drop across R1 during the sample time of the ADC.  This will cause the sporadic measurements you are seeing.

Although you will require a capacitor at C14, the value you list of 0.1uF is quite large for the purpose of transferring charge.  You could probably get away with 100 to 1000pF.  This would allow you to turn on and off the circuit much quicker.

Here's a link to an app note that helps clarify this.  See section 7.
ADC App Note