Like you already suspect a DMM won't give you necessarily the answer. If the only levels you read are only 3.3V (capital "V", since it's derived from a person's name) and 0V it's a safe bet that it's digital. A PWM signal is also digital, but will show a voltage between 3.3V and 0V on a DMM, since it includes a low-pass filter. The DMM won't tell an analog and PWM signal apart. You'll need an oscilloscope to actually see what the signal looks like.
If you can trace the signal back to the output of a microcontroller, without filtering, it's most likely a digital signal. While most microcontroller families are available with on-chip ADC, very few also have DACs on board.
There are several factors here.
First, the input impedance of the ADC. The ATmega328P uses a Successive approximation ADC. As such, the input is basically the input to a comparator, so the ADC has very high input impedance.
The ADC is specified as having a 100 MΩ (that is MegaOhm) input impedance.
However, this seems somewhat suspicious to me. Together with the fact that there is no analog input leakage specified, I would guess that this is the electrical characteristics of just the ADC, rather then the ADC together with the entire IO pin structure. I would guess that the ADC IO lines that are shared with digital IO have much more leakage current (1 uA from the docs) then the IO lines that are analog-only (50 nA, assuming that the SAR comparator is similar to the analog comparator input topology).
However, there is another consideration here, which is the reason that Atmel specifies the < 10 KΩ source impedance:
Basically, the input connections to the ADC inside the chip, after the multiplexer have some capacitance. If you look at the equivalent circuit for the ATmega ADC input:
You can see what the input looks like.
The problem with high source impedances arises when you are switching the input multiplexer from one pin to another. If you have two inputs, one at 0.5V and one at 4.5V, when you switch from one to the other, the input has to charge (or discharge) that 14 pF capacitor.
If the signal source is very high impedance, having to charge the capacitor may cause the input voltage to drop temporarily. If the ADC converts on the input while is is still charging the capacitor, you will get an incorrect value.
This can probably be dealt with by letting the ADC input settle for a period of time after switching ADC channels, but the best way to deal with it is to simply ensure that the input source can charge the capacitance fast enough that it's not a problem.
Yes, the analog pins on the Arduino can be used as digital outputs.
This is documented in the Arduino input pins documentation, in the Pin Mapping section: