(I'll assume that in a general model category C, Cyl(X) really means: a factorization of $A\to X$ into a cofibration $A\to \mathrm{Cyl}(X)$ followed by a weak equivalence $\mathrm{Cyl}(X)\to X$.

A sufficient condition on objects for the map in question 1 to be a weak equivalence, is that all the objects X,Y,A be *cofibrant*.

A sufficient condition on C for the map in question 1 to be a weak equivalence, is that the model category be *left proper*.

It's an interesting fact that in Top, the this also works if Cyl(X) denotes the "classical" mapping cylinder construction (X union a cylinder on A), which isn't necessarily a cofibration in the Quillen model structure.  The slickest proof is to use the "excisive triad theorem", as proved by May in *A Consise Course in Algebraic Topology*, p.79.  This also leads to a proof that Top is left proper.