The mobility of a series of linear alkanes in their inclusion compound with tris(o-phenylenedioxy)spirotriphosphazene is studied by high-resolution carbon-proton magic-angle spinning solid-state NMR spectroscopy. Two different carbon-proton dipolar recoupling experiments are compared with respect to their ability to yield precise site-specific, motion-averaged dipolar coupling constants. The most accurate results are obtained by analysis of extrema positions in Lee-Goldburg cross-polarization build-up curves. We present a comprehensive collection of coupling constants, which evidence a rotational motion of the all-trans chains around the channel axis, with some further averaging due to additional fluctuations, as previously found for alkanes in other host matrices such as urea. The order parameter increases toward the inner parts of the chains, and is largely independent of chain length. Notably, chains in a TPP host are not more ordered than in urea, even though the average TPP channel diameter is reported to be smaller. Significantly decreased order is found for highly filled short-alkane samples, which is interpreted in terms of an increased rate of mutual collisions. From residual dipolar couplings as well as carbon chemical shifts, we derive similar amounts of gauche conformers. Translational motions along the channels are further studied by proton double-quantum spectroscopy, which probes guest-host dipolar couplings. The extent of local-scale lateral motion is again correlated with the sample filling, and is a weak function of temperature, as expected from a case in which highly restricted single-file diffusion should dominate the mobility. Characteristic effects of sample aging are apparent in all our experiments.