The question is exactly that in the title: is there a forcing which collapses $\omega_1$ to $\omega$ but does not add a dominating real ("real" here meaning "element of $\omega^\omega$")?
It seems like the answer should be "no," and I've attempted to prove this myself. The problem is that the easiest way to do so would be to define a dominating function in terms of an arbitrary surjection $f: \omega\twoheadrightarrow\omega_1;$ however, there seems to be no clear way to do this. My first thought was to look at the set $S_f=\lbrace n: \forall m(m<n\implies f(m)<f(n) \rbrace$. This is certainly a real, but there is no reason it should be dominating, let alone not present in the original model already; in fact, we can alter the usual collapsing poset to demand that $S_f$ be precisely the evens, or precisely the powers of 17, or in fact any infinite co-infinite subset of $\omega$.
Hence, my question. Additionally, I have a meta-question: I am not a set theorist, so when I think about set theory I have a hard time judging which of the questions that occur to me are trivial and which are actually hard. In particular, I have placed this question here (as opposed to at MO) because I strongly suspect it is not actually very hard at all. My meta-question is, "Would this question have been appropriate at MO?"