When listening in to an AM signals of various frequencies, how do we exactly tune in?
Ok, I think here is a good point to introduce equivalent baseband, because you implicitly are already using it! 

So, what does your cosine 5Hz audio signal look like, if you were to set $f_c=0$, in spectrum?

Exactly, one dirac at +5 Hz, and one at -5 Hz! Hence, when you mix that up to $f_c\ne 0$, you get symmetric sidebands. That works for any real signals (ie. for any audio signal) – and even when you add them. That's where your two sidebands come from – positive and (hermitian) symmetrical negative components of any real-valued signal.

So, in baseband, your sum of 5 and 30 Hz cosines have four spectral components – at -30, -5, +5 and +30 Hz, and mixed up to the carrier frequency you get the same discrete spectral components, but added to $\pm f_c$. (your figure also shows a component at $f_c$, but that's only there if you got a DC component in your baseband signal, and/or you're not suppressing the carrier)

Now, there's very different receiver architectures for AM-modulated audio. None of them "just looks every 1/(audio period)". "Looking every so often" presumes you're building something digital. Most AM receivers (most AM being pretty obsolete by now) are not digital. The simplest detector methods really are just rectifying diodes and a low pass filter to get rid of the RF content – I'll leave googling for "AM diode detector" up to you; there's a plethora of good articles on that out there. This is all continuous-time, so there's no "looking that often" there – it all happens by electronically processing the continuous signal.

Now, assuming we're really aiming for digital here:

That 1/(audio period) wouldn't even make sense for audio – it breaks Nyquist; you need at least twice the audio bandwidth as sampling rate to be able to reconstruct the signal.

What one can build is simply a mixer with $f_c$, which will first (continuous-time!) multiply with a harmonic of that frequency (effectively, a complex sinusoid), and then sample (that's the act of looking only every so often) that, and then you get a digital signal that's basically nothing but your original audio sum. 

Note that you've been asking about "how to deal with the fact that you add two sines of different frequencies", but that question is totally independent from the AM aspect of that: you just can. Under the Fourier Transform, i.e. in the spectrum, these are orthogonal, i.e. you can perfectly separate them, both in a computer, in a circuit, or with your ear. That's why you're hopefully able to listen to sounds that aren't made of a single tone. In fact, in reality, single-tone sounds are extremely rare.