How to make an Audio Amplifier
Yes, but not with that circuit set up. You'd need a different one, and large capacitors. Edit: More explaining is necessary. The circuit you have provided is a bridged topology for only one channel. You can either double this circuit and have one copy drive each channel separately OR you can use one amplifier with the circuit at the top of the data sheet you yourself have provided (and hopefully read). That circuit in the datasheet is a single ended configuration. The Large capacitor is what would sit between the amplifier and the speaker so that you don't have a DC short from 12V (mid-rail) to ground. If you want a single ended amplifier without a capacitor then you need a bi-polar supply.
Ground planes are generally bad for audio amps, introduces too much coupling. I prefer routing the power and ground directly under each other to maximize the capacitance on the lines for high frequency AC shunting. Also, most importantly, star ground. Edit: Star ground information can be found very easily by using the power of the internet. A brief explanation: Current flows through ground just like all wires. Large currents, such as those experienced by power amplifiers, will cause a voltage drop on the ground wires enough so that the "ground" connection of different components will not be at the same potential. This can cause all sort of inaccuracies. But seriously, there are a thousand and one examples elsewhere.
It doesn't hurt you per say to not have them, but if you have a scope with an fft and can find the harmonic frequency you can filter better. Also, since that's a chip, it should have been made with some PSSR (power supply rejection ratio) look that up and figure out how much the noise from the power supply will get in. Edit Just looked at datasheet. Yes you want to filter that, there's no frequency/rejection chart. Who knows what will make it in, but 50dB is not great.
Vadj, lets you adjust the voltage output up and down from 24V. Not too far either direction, but enough that you can compensate for load variations and such (or run at 26V if that's what you wanted, with a drop in current capacity of course [it's a power supply, i.e. you get 48W no matter what voltage it gives]).