Is convolution distributive over multiplication?
Addition is (by nature) additive. Binary operations like multiply and convolution are both somehow of multiplicative structure. Thus, I suspect there is no generic distributivity (apart for Boole or Boolean algebras, where the restriction on allowed values shrinks the problem, see below). 

However, there is a sort of commutativity of multiply and convolution, as follows. It is interesting to first look at such operator  properties from an algebraic point of view. Properties are for instance, for a generic operator binary $\bigcirc$: 


commutativity: $a \bigcirc b = b \bigcirc a $
associativity: $a \bigcirc( b \bigcirc c) =  (a \bigcirc b) \bigcirc c  $
alternativity: $a \bigcirc( a \bigcirc b) =  (a \bigcirc a) \bigcirc b  $ or $(a \bigcirc  b) \bigcirc b =   a \bigcirc (b \bigcirc b)$
flexibility: $(a \bigcirc  b) \bigcirc a =   a \bigcirc (b \bigcirc a)$


Distributivity of a binary   operator $\diamond$  over operator $\bigcirc$  is more involved: it can be left-distributed: $$a \diamond (b \bigcirc c) = (a \diamond b) \bigcirc (a \diamond c)$$ or right-distributed: $$(a \bigcirc b) \diamond c = (a \diamond c)  \bigcirc (b\diamond c)$$ 

Division is right-distributive over add, not left-distributive. Distributivity of multiply over add is at play in rings and fields. But the converse does not apply. There are mathematical structures where 
two binary operators distribute over each other, like Boolean or switching algebras. Apart from that, I cannot remember of standard cases of multiply-like $\ast$  or $\cdot$ that would distribute. However, I will details some specific cases, useful to signal processing, ending with the Hilbert transform and Bedrosian theorem.

In the case of multiplication and convolution, or $a$ scalar, this is associative, as : 

$$a.[b(t)\cdot c(t)] = [a.b(t)]\cdot c(t)= b(t)\cdot [a.c(t)]$$
and
$$a.[b(t)\ast c(t)] = [a.b(t)]\ast c(t)= b(t)\ast [a.c(t)]$$

You see that you cannot directly"distribute it", as this would become "bilinear".
Convolution and point-wise product are linear. So, if you consider that $c(t)$ (or  $b(t)$ by symmetry), there exists a linear operator $\Lambda$ such that:

$$\Lambda(c(t)) = a(t)\ast[b(t)\cdot c(t)] $$

Whether $\Lambda$ has a simple closed form expression is a difficult matter. As a side note, where $\mathcal{H} $ denotes the Hilbert transform, the Bedrosian theorem (A product theorem for Hilbert transforms states that (under some conditions):

$$\mathcal{H}(a(t)e^{i\theta t}) = a(t)(\mathcal{H}(e^{i\theta t}))\,.$$

where $a(t)$ is an (low-pass) envelope, and $e^{i\theta t}$ a modulation. More generally, if a low-pass signal $x_\flat$ and a high-pass  $x_\sharp$ have non-overlapping spectra, then under Bedrosian theorem:

$$\mathcal{H}(x_\flat(t)\cdot  x_\sharp(t))=x_\flat(t)\cdot \mathcal{H}(x_\sharp(t))$$

Now, let us remind that the Hilbert transform can be seen as the convolution with the distribution ($\operatorname {p.v.}$ is the Cauchy principal value):

$$h(t) =\operatorname {p.v.} {\frac {1}{\pi t}}$$

Thus:

$$h(t) \ast (x_\flat(t)\cdot x_\sharp(t))  =  x_\flat(t) \cdot(h(t)\ast x_\sharp(t))$$

This is the only practical case I can remember of a switching of operators in signal processing, more a form of limited commutativity.