Alkali metal adsorbates induce a 3x1 reconstruction of Si(111) widely believed to be a "honeycomb-chain channel" structure. This model is based on a true Si=Si double bond in the surface layer, which at 1/3 ML coverage leads to the elimination of all dangling bonds and thus keeps the fundamental gap free of surface states. Alkaline-earth adsorbates also appear to induce 3x1 (as well as higher-order) reconstructions, yet despite the extra electron the resulting surfaces remain fully gapped---an apparent contradiction to the one-electron band picture. Lee et al. recently suggested that for alkaline-earth adsorbates the coverage is in fact 1/6 ML, and showed that a 3x2 honeycomb-chain channel model is indeed fully gapped.@footnte 1@ We use density-functional methods first to confirm that the model of Lee et al. is indeed energetically preferred at low coverages. Second, we predict that structurally related 5x2 and 2x1 reconstructions will appear---if they are not preempted---at higher coverages. Finally, we propose an explanation for why the 1/6 ML phase appears 3x2 in STM but 3x1 in LEED: namely, that at moderate temperatures the adsorbates will exhibit only short-range order, due the near energetic degeneracy of H3 and T4 adsorption sites. To demonstrate this, we extract from our density-functional calculations the adsorbate-substrate and adsorbate-adsorbate interactions, and based on these carry out classical Monte Carlo simulations to explore the detailed temperature dependence of adsorbate ordering. @FootnoteText@@footnte 1@G. Lee, D. Shin, H. Kim, J. Koo, and S. Hong, Bull. Amer. Phys. Soc. 46 (2001).