Is whatever we work out about that frame of reference (e.g. his
distance from Bohr after 3 seconds) only true for Planck, or is it
also true for us, who are not in that frame of reference?
Where v is the relative velocity between the observer and the moving object, and contraction is seen along the line of motion. So distance measurements can depend on relative motion.
What you need is the concept of invariance in physics. A good example is the speed of light in a vacuum, which is the same in all inertial (non-accelerating) frames of reference. This invariant speed doesn't cause conflicts with our experiences & our intuitions, until we approach speeds of the order of magnitude of light. When that happens, we find time isn't the same for all observers, and because of that we need transformations between different observers, that account for their changed clocks.
What has happened, is we had assumed a spacial symmetry, that speed looks the same for all observers plus or minus theirs in a linear way; and a time symmetry, that events can all be related to one clock regardless of speed. Then, we found at high speeds these linear symmetries-under-transformation between observer viewpoints, fail, and turn out to be linked, and high speeds relative to an observer cause some of the dimensions to swap so their combination keeps certain qualities, rather than each dimension we measure. There isn't space, and time, instead there is space-time. The space-time view is the same for all observers, but space & time may shift in related ways. This explorable explanation helps get an intuition for how things squash and rotate, and how event information moves.
Accelerated frames, involve forces, which means varying energy, requiring the energy momentum relation, which means general relativity. Then our intuitions become even less useful.
We thought we had some other separate things we can observe about a system, time-reversability, charge, and parity ('handedness'). It turns out these are also linked, in CPT symmetry, meaning one can be violated if one or more of the others is also, in a special way such that a combination of the three is still conserved.
Continuous symmetries-under-transformation and conservation laws, are related by Noether's theorem. Translational symmetry and conservation of momentum. Time symmetry and conservation of energy. Using this we can start to understand how time & space may not be 'out there', but be patterns of symmetry 'in' each local point. Quantum gravity theories like Loop Quantum Gravity build the dimensional symmetries from something simpler, a spin network of quantum information. It's not the only option, but space-time is expected not to be continuous, and must mesh somehow with wider quantum symmetries like CPT.