Regarding Questions 1 and 3, it's not crucial that $g$ be non-vanishing in $X$; after all, $x$ (i.e., the identity function) vanishes. However, it is important that $g$ be locally one-to-one (and therefore strictly monotone if $g$ is continuous and the domain $X$ is an interval) in order for $df/dg$ to be defined throughout $X$.
The answer to Question 3 is "no": Take $f(x) = 1 + x^3$ and $g(x) = 1 + x^5$ on $X = (-1, 1)$. ($dg/df(0) = 0$ and $df/dg(0)$ is undefined. The added "$1$"s guarantee $f$ and $g$ are non-vanishing, but have no effect on the difference quotient for $df/dg$.)
Here's are some (variously far-flung) musings as to why this definition of $df/dg$ is not widespread.
A derivative $df/dx$ may be viewed as measuring the rate of change of $y = f(x)$ against a uniform standard rate, namely $1 = dx/dx$.
To compare the relative rates of change of $y = f(x)$ and $z = g(x)$ (when this makes sense), one is naturally led to compute a ratio of derivatives: As Arthur notes, if $f'(a) = df/dx(a)$ and $g'(a) = dg/dx(a)$ exist for some $a$, and if $g'(a) \neq 0$, then $df/dg(a)$ exists and is equal to the ratio $f'(a)/g'(a)$. (These observations permeate the discussion so far; just spelling them out explicitly.)
One philosophical point in the two preceding paragraphs is the analogy to the problem of converting data (say, image or audio files) between multiple formats: It's easiest to create a single master format $M$, and to convert format $A$ to format $B$ by converting $A$ to $M$, then $M$ to $B$.
In this analogy, $y = f(x)$ is format $A$, $z = g(x)$ is format $B$, and $x$ is the master format $M$; "converting" $A$ to $B$ means "taking the rate of change", and breaking up the conversion corresponds to the formula $df/dg = (df/dx)/(dg/dx)$.
To complete this train of thought, "$df/dg$" can be accommodated (modulo details, see below) within the framework of differential calculus.
As to precise hypotheses: If the domain $X$ is an interval and if $f$, $g$, and $h$ are differentiable with non-vanishing derivatives, then the answer to all three questions is "yes". However, these are restrictive hypotheses, and they appear to be "(close to) minimal" if one wants all three questions to have an affirmative answer.
Of course, these hypotheses are not necessary if one merely wants $df/dg$ to be defined, but if $h$ is differentiable and $f = h\circ g$, then existence of $df/dg$ implies relatively little about $g$. In an extreme case, $h(x) = x$, $g$ is injective (but otherwise as pathological as you like), and the difference quotient for $df/dg$ is identically equal to $1$.
That is, the proposed definition of $df/dg$ "works entirely as expected" only under restrictive hypotheses (that are already handled well by differential calculus), but in general is too lax to put any substantial restrictions on $g$.