Can there exist a fractional linear transformation mapping the upper half-disc conformally onto the upper half-plane? I think there cannot be, because I am looking for where the unit circle and the real line (whose union is a superset of the boundary of UHD) would be mapped. Since the boundary of $UHP$ is a single line, it seems there's no way it can work.
Side question: how can I make rigorous the notion that FLTs preserve boundaries? I remember something about this from my complex analysis course, but I'm forgetting where I should look for it.
Your logic is correct. There can not exist a FLT mapping the upper half-disc conformally to the upper half plane. FLTs take circles to circles (or lines through the origin), and such a map would necessarily map the boundary of the upper half disc (not a circle) to the boundary of UHP (which is a circle).
It is a theorem that every FLT of the UHP extends uniquely to a map on the boundary $\mathbb{R}\cup\{\infty\}\cong S^1$. This theorem (along with the Brower fixed point theorem) is used to show that every fractional linear transformation is either hyperbolic, parabolic, or elliptic. You can find a proof in any introductory hyperbolic geometry textbook (e.g., Anderson's Hyperbolic Geometry, or Marden Outer Circles for a more advanced perspective).