Typically, numerical simulations of Earth systems are coarse, and Earth observations are sparse and gappy. We apply four generative diffusion modeling approaches to super-resolution and inference of forced two-dimensional quasi-geostrophic turbulence on the beta-plane from coarse, sparse, and gappy observations. Two guided approaches minimally adapt a pre-trained unconditional model: SDEdit modifies the initial condition, and Diffusion Posterior Sampling (DPS) modifies the reverse diffusion process score. Two conditional approaches, a vanilla variant and classifier-free guidance, require training with paired high-resolution and observation data. We consider multiple test cases spanning: two regimes, eddy and anisotropic-jet turbulence; two Reynolds numbers, 10³ and 10⁴; and two observation types, 4x coarse-resolution fields and coarse, sparse and gappy observations. Our comprehensive skill metrics include norms of the reconstructed vorticity fields, turbulence statistical quantities, and quantifications of the super-resolved probabilistic ensembles and their errors. We also study the sensitivity to tuning parameters such as guidance strength. Results show that the generated super-resolution fields of SDEdit are unphysical, while those of DPS are reasonable but with smoothed fine-scale features; however, neither of these lower-cost models propagates observational information effectively to unobserved regions. The two conditional models require re-training, but reconstruct missing fine-scale features, are cycle-consistent with observations, and predict correct turbulence statistics, including the tails. Further, their mean errors are highly correlated with and predictable from their ensemble standard deviations. Results highlight the tradeoffs between ease of implementation, fidelity (sharpness), and cycle-consistency of the diffusion models, and offer practical guidance for deployment.