why do I get noise on the output of the LM386 when I connect speaker but not when I connect a resistor?
One aspect of the question:


  Why do we have the output connected to ground via a 0.05uF capacitor and a 10ohm resistor in most schematics in the datasheet? What purpose does this serve since this is clearly not a filter.


This is the "Zobel network". It is for stability against oscillation; it provides a load for the amplifier at high frequencies (which the speaker doesn't, being an inductor). This feature is very commonly seen in the designs of audio power amplifier output stages.


  This is clearly a single supply amplifier topology. However, even if I provide it with a 100mV p-p signal, the negative cycle still shows at the output. How come?


Though single supply, it is internally biased. The thing to remember is that this is not an OP amp, even though the schematic symbol closely resembles one. It helps to look at the representative schematic of the internals that is shown in some of the data sheets. You will see certain features, like 50 \$\text{k}\Omega\$ biasing resistors on both inputs. These generate a voltage when the bias current flows through them out of the transistor bases. Note how the speaker load is coupled with a big capacitor; that's to remove the DC offset on the output. 


  Why is output always capacatively coupled to a speaker? Is it because the DC signal may overheat the speaker?


That's basically it. But not only overheat, but DC through a speaker generates a magnetic field in its coil, which reacts against the speaker magnet, producing a mechanical displacement of the speaker cone. That could cause distortion, and lowered headroom.