Short answer without reference to "first order logic" or "modal logic": Even $P$ and $Q$ are valid propositions in the appropriate context, hence there is no reason why your two sentences shouldn't be valid propositions. Note that I interpret "valid proposition" in the sense of "admissible proposition". You wonder about the value of $x$ here. The typical context in "first order logic" is that $x$ has a fixed value, but this often confuses people (because such a proposition is rarely useful in isolation). In other contexts like equational reasoning, it's also possible that $x$ is implicitly universally qualified (i.e. the proposition makes a statement about all possible $x$).
In the appropriate context, both of your sentences can be valid propositions. Let's start with $x^2 = 1$, because it's already stated in a "sugared" formal language. Let's assume its "desugared" formula to be $x\cdot x = 1$. Then it will be a valid proposition for a first-order language containing a constant "$1$" and a binary operator "$\cdot$". Here I interpret "valid proposition" as being member of the corresponding "formal language", i.e. in the sense of "admissible proposition". So the proposition could be false for some interpretation, but it would still be valid.
You wonder about the value of $x$ here. In the context of first-order logic, the value of $x$ is fixed as part of the interpretation. In the context of equational reasoning or universal algebra on the other hand, the equation (we don't usually talk about propositions in this context, only about equations and clauses) is only true (sadly, the correct word in this context would be "valid") if it is true for all possible values of $x$. Anyway, the point is that the formal semantics often allows free variables, and takes care to assign them an appropriate meaning in the interpretation.
Your second sentence "Today is Thursday" would only be used as an example for a proposition in the context of modal logic. In modal logic, we have both an interpretation and a collection of worlds, and we evaluate the truth of a proposition for a specific world. So the interpretation would give meaning to the words "Today", "Thursday", as well as to the connective "X is Y". The possible dates would also be fixed by the interpretation as the possible worlds (and how these possible worlds are related to each other by the modal operators), but the actual date/world where you evaluate this proposition is a part of the evaluation context, and not part of the interpretation. And the propositions are true or false with respect to a specific evaluation context, which includes and an interpretation and a specific world.
One might imagine designating a specific world as the actual world as part of the interpretation, but I don't know whether this is commonly done. From the texts on model logic I read so far, I have seen it once or twice, but it seems more common not to do this.