Interaction between Analog Pins when measuring a DC voltage- a common issue?
This is normal and expected behaviour.  There are a number of factors that influence it, and it will help to understand the basic internals of the ADC.

In simple terms the front-end of the ADC looks like this:







simulate this circuit &ndash; Schematic created using CircuitLab

The MUX selects which pin (A0-A5) is being sampled. Switch SW1 connects the Sample And Hold capacitor C1 to that selected input, which then charges (or discharges) to the same voltage as the input.

After the acquisition period switch SW1 switches position and the ADC circuit is connected to the capacitor.  Over a period of clock cycles the ADC works out what the voltage across the capacitor is and stores the result in the ADCH and ADCL registers.

Now - on a pin that has nothing connected to it, what voltage will be across that capacitor when SW1 is switched to charge it? Well, this is where your strange results come from:


Any charge already in the capacitor will still be there (which is why it can be good to connect the unused ADC pins to ground through a resistor, to remove that charge).
Signals, either in wires, IC pins, or the silicon circuitry in the chip itself, will be capacitively coupled with what is next to it.


That last one is the most critical for your results.  Capacitive coupling.

Place two wires next to each other and they will, through capacitance, try and get the same charge on them. Under most circumstances that attempt to influence the charge is completely swamped out by what is being send down the wire. But, when there is nothing being sent down the wire that charge becomes noticable. A1 will attempt to "float" to the same level as A0, because they're next to each other. A2 will try and float to the same level as A1, A3 to A2, etc. 

If you put a signal in to A0 and plot all 6 inputs you will see each has a progressively lower amplitude copy of the signal. It's hopping, capacitively, between the pins. And all because there's nothing to stop it. 

If you think "Ah, but it's DC. DC doesn't go through capacitors", then you're wrong.  Wrong in thinking it's DC.  There's a lot of mistaken assumptions about the labels "AC" and "DC".

DC is not what you have.  Yes, there is a "DC component" to what you have (which is manifest as the average of the voltage over time), but there is a large AC component as well.  That AC component is what you are graphing in the form of your changing battery voltage. 

Voltage is far more complex than just DC or AC. What you think of as AC is actually a waveform with a fundamental frequency (e.g., 60Hz) and zero "DC offset". Add a DC offset to the signal and it's still AC, but compared to the "lower" end of whatever adds the DC offset, it's just the waveform is "shifted up".

Pure DC, and what doesn't pass through a capacitor, has no signal components. There is only 0Hz and nothing else. Pure DC is almost impossible to create without highly specialized equipment. All voltages in normal life vary, and that variance is the AC component of the signal.