Stability of cyclohexane type species
According to the Ref.1, 5-hydroxy-1,3-dioxane (the compound of structures V and VI) both in the gaseous state and in dilute $\ce{CCl4}$ solution exists as a chair conformer, the hydroxy group in an axial position with an intramolecular hydrogen bond of the $\ce{O—H⋯O}$ type:

Accordingly, the axial conformer (V) is more stable by $\Delta G^\circ = \pu{-1.2 kcal mol−1}$). Microwave spectroscopy and the $^3J_\ce{H(5eq),OH}$ coupling constant suggest that the $\ce{OH}$ group lies in the plane of symmetry $(\ce{C_{(2)}-C_{(5)}-O}\text{-plane})$ and is a part of a bifurcated hydrogen bond to the two ring oxygen atoms (Ref.2). Therefore, it is safe to say the conformer V is more stable than conformer VI between two possible conformers of 5-hydroxy-1,3-dioxane (marked with $\color{green}{\text{green}}$ oval).
It is well known that the most stable conformer of tert-butylcyclohexane is tert-butyl in equatorial position. In a 1,4-substituted version of 1-tert-butyl-4-methylcyclohexane, since methyl group is so small compared to tert-butyl group that I is almost exclusive in equilibrium (marked with $\color{green}{\text{green}}$ oval).
However, I wasn't so sure about stability between III and IV. There is clearly no H-bonding opportunities except theoretical chemist may argue hyper conjugation with ring oxygen by MO calculations. I leave it open for any computational chemist to prove that possibility, yet my best guess is III more stable between them (marked with $\color{orange}{\text{orange}}$ oval with question mark).
References:

J. C. Jochims, Y. Kobayashi, “Bifurcated hydrogen bonds in z-2-phenyl-1,3-dioxan-5-ols,” Tetrahedron Letters 1976, 17(24), 2065-2068 (https://doi.org/10.1016/S0040-4039(00)93819-X).
Jose L. Alonso, E. Bright Wilson, “Study of an intramolecular, bifurcated hydrogen bond in 1,3-dioxan-5-ol by microwave spectroscopy,” J. Am. Chem. Soc. 1980, 102(4), 1248–1251 (https://doi.org/10.1021/ja00524a005).