I've now got a basic kinematics controller using a ghost object for collision testing, alongside (rather than instead of) the player-controlled rigid body mapped to the sphere. The sphere's still there, but now invisible.
Now what happens is that the player object exerts forces upon non-static objects it collides with, while static (or otherwise immovable) objects collide with the ghost and prevent movement. Unlike the day #6 video, you actually stop when you hit walls.
For reference, the different-coloured cubes and the other sphere each have different physical properties (mass, restitution, surface friction), and so behave differently. You can see that the red cubes have very low friction, for example, so they look like blocks of ice. Bullet does all of the heavy lifting there.
The first-person camera's also been better set up, with a near clipping distance reduced to the minimum so that you can actually see what you collide with. At the moment, the camera's positioned at the sphere centre (which actually works quite well), but it should be adjustable in the actor properties.