Problem translating an argument into predicate logic. The (clearly flawed) argument I am attempting to translate is this: "All my dreams are in black and white. Old TV shows are in black and white. Therefore, some of my dreams are old TV shows". This is my attempt:
$$\forall x (D_x \to B_x)$$
$$O_x \to B_x$$
$$\therefore \exists x(D_x \land O_x)$$
where $D_x:x$ is a dream, $B_x:x$ is in black and white, $O_x:x$ is an old TV show, and the domain/universe of discourse is "all my dreams and old TV shows".
My first concern is whether it would be more suitable to have $\forall x(O_x \to  B_x)$ as the second line, since the English could be rephrased to say "all old TV shows are in black and white"?
Secondly, should my domain/universe of discourse just be "dreams and TV shows"? It just seems oddly specific.
 A: The domain "all my dreams and all TV shows" is OK.  Concerning the other question, yes, you should universally quantify $x$ in $O_x \rightarrow B_x$.  Otherwise you are left with a free variable and the formula is not a sentence.  That is, its truth value depends on the value of $x$.
Finally, if the conclusion follows from the premises, the conjunction of the premises and the negation of the conclusion is unsatisfiable.
However, if you have two distinct elements in your domain, one is a black-and-white dream of yours and the other is an old black-and-white TV show, then you can satisfy the premises (all dreams are B&W and so are all TW shows) while also satisfying the negation of the conclusion. (No element of the domain is both a dream and a TV show.)
Note that formally you can use $x$ in both premises, but you are better off using a different variable in the conclusion: $\exists y (D_y \wedge O_y)$. It's easier to avoid confusion, even though in the end you'll be checking for the satisfiability of
$$ \forall x ((D_x \rightarrow B_x) \wedge (O_x \rightarrow B_x) \wedge (\neg D_x \vee \neg O_x)) \enspace. $$
You may wonder why is it OK to have just one dream and one TV show in the domain.  The explanation is this: if there is one structure in which the premises are true, but the conclusion is false, then the conclusion does not logically follow from the premises.
In another structure (you may say, in an alternate world) both premises and conclusion may hold, but the existence of a structure like the one above says that there is no logical consequence.
