Help figuring out an op amp circuit with two loudspeaker outputs
If you redraw your schematic using the normal triangle op-amp symbols it will be easier to understand your circuit.



simulate this circuit &ndash; Schematic created using CircuitLab
Figure 1. Upper circuit redrawn.
The idea of a schematic diagram is to show the schema of the circuit. Yours is a very bad example as it it quite difficult to identify the various parts. In Figure 1 we can see clearly the power supply the voltage divider and the inverting op-amp.

.. at Pin 3 (non-inverting input) there should be 1.6 V.

Correct. Because you have a single supply we need to bias the audio signal mid-way between V+ and GND.

I read that through a feedback loop the voltage at both inputs (inverting/Pin 2 and non-inverting/Pin 3) should be the same (1.6 V). Therefore there should be a gain of 11 (with R3 and R4) at output 1 right?

No, for an inverting amplifier the gain is given by \$ A = - \frac {R_f}{R_i} = -10 \$.

I'm slightly more confused at what's going on at pin 5-7 ...

The second amplifier is a non-inverting amplifier. Its gain is given by \$ A = 1 + \frac {R_f}{R_i} = 1 +\frac {90k}{10k} = 10 \$.
The idea here is that since your supply voltage is so low we drive the speaker in &quot;bridge&quot; mode. While one output goes up, the other goes down and vice versa. Now you will get double the voltage on your speaker and four times the power (since \$ P = \frac {V^2}R \$.

I'm thinking of switching the 3.3 V input to 5 V because of the loudspeakers high impedance of 50 ohms, so it can get more power.

Good idea.

Fault finding
With no music signal the output of U1A and U1B should be the same voltage as the junction of R5-R6.
C3 and C4 are too small and you will get no bass. Try 100 μF or something around that size.

About your title:

Help figuring out an op amp circuit with two loudspeaker inputs

The diagram shows two outputs to one loudspeaker.

From the comments: What do the capacitors do?

Figure 2. OP's redrawn circuit.

R5 and R6 are the mid-supply reference voltage, discussed above. C5 holds that voltage steady by filtering out any noise on the 3.3 V supply.
C1 is DC blocking. It allows the audio signal in but prevents R4 being grounded by whatever is driving the input. e.g. A guitar pickup is just a coil of wire and would provide a DC path which would affect the outputs.
C2 does the same job for U1B and prevents its DC operating point (remember that you want the music signal to oscillate around 3.3/2 V) from being upset.
C3 and C4 remove the 3.3/2 V DC from the output. These would normally be there for audio outputs to another stage. Since you are connecting your load across the two outputs you can omit those with very little chance of damage. When the input is silent those two outputs should be at the same 3.3/2 V and therefore no current will flow in your &quot;speaker&quot;. This is done in car audio systems, for example, and results in a much better bass response as there are no capacitors (which would act as high-pass filters otherwise).