(1) Does 'input 2' correspond to the +12V wire and 'input 1' to the -12V wire?
NO. This is connected to the power supply connector. (Pin 1 is the positive, pin 2 the ground or 0V). The INPUT connector is where the audio signal goes (2) is the live and (1) is the ground.
(2) When I see the symbol 'ground' on the schematic does that mean that I need to connect those to the -12V line? I have no 0 volt line in the circuit so what is the ground line?
All voltages are relative to each other. If you only have TWO wires from the AC/DC converter then the most positive is taken as the positive and the other one is the OV or ground. A simple check with a voltmeter will determine which is which.
(3) Why are C3, C5 non-polarized capacitors and why are the other ones polarized? The current flows from negative to positive. Does that mean that the polarized capacitors need to be placed with their positive legs pointing to the left?
C3 and C5 are small value capacitors (0.1uF). These can be easily made as physically small, non-electrolytic types. They have the advantage of being able to decouple (short out) the higher frequencies. The larger values (uFs) are made as electrolytics as these can be made with high values in small physical packages. They are generally much poorer at handling the high frequencies. By combining an electrolytic with a non-electrolytic capacitor in parallel (eg C3, C6) you get a much better response over a wider range of frequencies. In this case they are used for 'smoothing' the supply voltage, preventing hum and hiss. The positive plate of an electrolytic is shown as an open rectangle but left and right (or up, down) have no meaning in terms of connection as this will be determined by board layout. Conventionally current is taken to flow positive to negative.
If you don't connect the grounds of the two supplies together, you don't have a circuit.
Assume for a moment that you don't connect the two grounds together. You connect the microcontroller's data out pin to the LED strip's data in pin. The LED strip determines if your microcontroller is sending a 1 or a 0 by measuring the voltage on the data pin compared to its own ground voltage - but since its ground isn't the same as the microcontroller's ground, it could get nearly any result. At best, the two devices won't be able to communicate, and at worst the voltages will be so wildly different that one device or the other will see a very high or very low voltage on the data pin.
Connecting the two grounds together means you have a circuit; everything is working from the same ground reference, and power can flow from either supply, through a device or devices, and back into the supply via the shared ground.
You're right that connecting grounds together can in certain cases cause a problem; this happens when both devices are already referenced to different ground voltages. For instance, one might be connected to a local power supply, while the other is connected over a long cable to a power supply a long way away on a different AC circuit; there's nothing preventing the two having different ground potentials, creating what's called a ground loop. There are various ways of isolating two systems but still transmitting data between them in situations where this can happen.
In practice, however, either one of your grounds will be floating - that is, having no external reference - because it's running off batteries or otherwise has no other ground connection, or both your grounds will ultimately be referenced to the same voltage; for instance, when you have two devices both plugged into AC power from the same distribution board. In such situations, connecting the grounds together is both safe and necessary to complete a circuit.
Best Answer
In most electronics and electrical circuits, "Ground" or "GND" is just the point in the circuit that we choose to call "zero volts", and use as a reference when measuring voltages elsewhere in the circuit - it is where we put the black meter lead.
Very often, "GND" is the most negative terminal of the power supply, but if the designer chooses, may be the positive terminal, or perhaps the mid-point of the power supply.
There is no "GND" part, and "Ground" has no special or magical properties.
In AC power distribution, and in some radio antenna systems, "Ground" does indicate a connection to the earth, but that clearly doesn't apply in portable battery-operated circuits.