Is it hypothetically possible for a planet to be both tidally locked, and still have a rotation "horizontally"? Where the substellar point would in effect be like the point of a top, spinning and yet also always pointing inwards toward the star?
Not really. The angular momentum of such a system is not conserved, as the rotation axis changes over time.
Simple-mindedly If a planet had, at some point in its orbit, its rotation axis pointing at its star, then that axis would remain pointing in the same direction (so not at the star, in general) throughout the orbit, and for almost all of the orbit there would therefore be tidal effects, which would slow the rotation.
In real life I imagine there would be seriously hairy interactions between the orbital motion and the planet's spin which would result in the axis wobbling all over the place and probably ending up in some more conventional direction in due course (the rotation axes of planets do change over time due to interactions like this).