The conditional PDF $g$ of $Z=Y-X$ conditionally on $X=x$ is such that, for every $z$, $g(z)=f(x,z+x)/f_X(x)$.
The conditional PDF $h$ of $T=X/Y$ conditionally on $Y=y$ is such that, for every $t$, $h(t)=|y|\,f(ty,y)/f_Y(y)$.
Which part of this causes you trouble?
Note: Formally, the conditional distribution of some $U$ conditionally on $V=v$ is some distribution, that is, a probability measure $\mu$. If $\mu$ has a density $m$, that is, if, for every $B$, $P(U\in B\mid V=v)=\mu(B)=\int\limits_Bm(u)\mathrm du$, then one says that $m$ is the conditional density of $U$ conditionally on $V=v$.
Edit: To compute the conditional PDF $h$, note that, by definition, for every measurable function $u$,
E[u(X,Y)\mid Y=y]=\int u(x,y)f(x,y)\mathrm dx/f_Y(y).
Using this for $u(x,y)=v(x/y)$ yields
E[v(T)\mid Y=y]=\int v(x/y)f(x,y)\mathrm dx/f_Y(y).
The change of variable $t=x/y$ yields $\mathrm dx=y\mathrm dt$ hence
E[v(T)\mid Y=y]=\int v(t)f(ty,y)(|y|\mathrm dt)/f_Y(y),
which yields the conditional density $h$.