Question about ASK, correlation, and frequency domain
You are mixing different notions.

What happens in basic digital transmission is :


Digital signal generation (0's and 1's)
Bit to symbol association (digital modulation), in our example ASK.
Shape filtering, which allows the signal's spectrum to be contained within a standardized shape, since we can't just throw a digital signal with very sharp transitions in the air, in would pollute the electromagnetic spectrum.
Finally, we mix the signal with a carrier frequency in order to shifts it's spectrum around the appropriate transmitting frequency


What happens at the receptor is the same operations in anti-chronological order :


Analog demodulation (the signal is mixed with a carrier of the same frequency)
Matched filtering, which is the same operation as in the emitter, but with a specific filter, which is ideally the complex conjugate and temporally reverted of the filter used in emission ($g^*(-t)$). Why is it this one and not another ? See Nyquist ISI criterion.
Symbol estimation
Digital demodulation of estimed symbol


About the filtering operation specifically : time domain and frequency domain operations are equivalent. If you do a time domain filtering, or any mathematical operation (under trivial notations) :
$$
y(t) = g(t)*x(t)
$$
It is equivalent to frequency domain multiplication :

$$
Y(f) = G(f)X(f)
$$

Analog amplitude modulation of a signal $r(t)$ is, like you say, the operation $r(t)\cos(2\pi f_0t)$ (multiplication) which in frequency domain is equivalent to $R(f)(1/2)*(\delta(f-f_0)+\delta(f+f_0))$ (convolution). 

Demodulation takes $r(t)\cos(2\pi f_0t)$ and multiply it again :
$$
s(t) = r(t)\cos(2\pi f_0t)\cos(2\pi f_0t)
$$
Which gives after simplifications :
$$
s(t) = r(t)\frac{1}{2}\left(1+\cos(4\pi f_0t)\right)
$$
That's when your LPF comes into action : it filters the unwanted $2f_0$ component and keeps only $r(t)$ which is the demodulated signal, and to do this mathematically, we write $s(t)*lpf(t)$, $lpf(t)$ being the LPF's impulse response.

In short : analog demodulation is either a multiplication in time domain ($r(t)\cos(2\pi f_0t)\cos(2\pi f_0t)$) followed by a convolution in time domain ($s(t)*lpf(t)$)

OR

It is a convolution in frequency domain ($R(f)*M(f)*M(f)$) (with $M(f)=(1/2)*(\delta(f-f_0)+\delta(f+f_0))$ followed by a multiplication in frequency domain ($S(f)LPF(f)$)

BUT both methods outputs are the same, physically speaking. We work in frequency or time domain according to which domain is best suited to give easy calculations, as you may have noticed.