A question concerning the hypothetical syllogism.

I am working on a proof and have found that given the statement if $$P(x)$$, then $$\lnot Q(x)$$ is true. Suppose that I do not know the truth value of if $$\lnot Q(x)$$, then $$R(x)$$. Taking the conjunction of these two statements allows for the conclusion if $$P(x)$$, then $$R(x)$$ by hypothetical syllogism. Now, assuming that it is known that if $$P(x)$$, then $$R(x)$$ is always false when $$P(x)$$ is true, and that a conjunction can only be true when both statements are true, does it follow that if $$\lnot Q(x)$$, then $$R(x)$$ is always false whenever $$\lnot Q(x)$$ is true? I am assuming that it is, but I want to makes sure there is not something I am missing where there is a case where if $$\lnot Q(x)$$ could be true, but also $$R(x)$$ could be true.

I am working on a proof and have found that given the statement if $$P(x)$$, then $$\lnot Q(x)$$ is true. Suppose that I do not know the truth value of if $$\lnot Q(x)$$, then $$R(x)$$. Taking the conjunction of these two statements allows for the conclusion if $$P(x)$$, then $$R(x)$$ by hypothetical syllogism.

That inference is valid.

$$P(x)\to\neg Q(x)~,~ \neg Q(x)\to R(x)~\vdash~ P(x)\to R(x)$$

Now, assuming that it is known that if $$P(x)$$, then $$R(x)$$ is always false when $$P(x)$$ is true, and that a conjunction can only be true when both statements are true, does it follow that if $$\lnot Q(x)$$, then $$R(x)$$ is always false whenever $$\lnot Q(x)$$ is true?

No, that is not entailed.

$$P(x)\to\neg(P(x)\to R(x)),P(x)\to\neg Q(x)~,~ \neg Q(x)\to R(x)~\nvDash \neg Q(x)\to\neg(\neg Q(x)\to R(x))$$

I am assuming that it is, but I want to makes sure there is not something I am missing where there is a case where if $$\lnot Q(x)$$ could be true, but also $$R(x)$$ could be true.

Yes, suppose $$\neg P(x)$$, $$\neg Q(x)$$, and $$R(x)$$. Well, all three premises are satisfied by this model, but the conclusion $$\neg Q(x)\to\neg(\neg Q(x)\to R(x))$$ is not satisfied. So the conclusion is not entailed by the premises.