If generating variables $Y | X$, then does one need to simulate independent $X$ for every $Y$ generated or is it enough to simulate $Y$s based on the same $X$?
If e.g. $Y | X \sim N(x,x^2)$ and $X \sim U(-1, 1)$, then can I simulate one $X=x$ and e.g. 100 $Y$s using the same $X$? Or do I have to create a new $X$ for every new $Y$?
If you use $X=x$ then you simulate from $Y|X=x$. So if you want to sample from the conditional distribution of $Y$ given random variable $X$, then you need different draws from $X$, because otherwise you'd ignore the variability of $X$. It makes a difference if you sample from $\mathcal{N}(0.5, 0.25)$ or from $\mathcal{N}(X, X^2)$, where $X$ is a random variable. But this is a very general answer, for a general question.
The conditional distribution of $Y$ conditional on $X=x$ is a distribution indexed by the value $x$: producing an iid sample$$(Y_1,\ldots,Y_n)$$ from this distribution is thus done for this value $x$. For instance, an iid sample from $\mathcal{N}(x,x^2)$.
Simulating from $Y|X$ as in a $\mathcal{N}(X,X^2)$ distribution does not really make sense in that what this means is simulating from a distribution with a random parameter, hence requires a simulation from $X$ prior to each simulation of $Y$ (as W. Huber pointed out), which amounts to simulating from $(X,Y)$ and taking $Y$ as a result. Thus "simulating from $Y|X$" actually means simulating $Y$ from its marginal distribution. Even though this may imply in practice simulating from $X$ first, as in the example of producing an iid sample $(X_1,\ldots,X_N)$ from a $\mathcal{U}(0,1)$ distribution, with value $(x_1,\ldots,x_N)$ then a second sample of $Y_i$'s, with $$Y_i\sim\mathcal{N}(x_i,x_1^2)$. (The $Y_i$'s are marginally iid.)
