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We present the KODIAQ-Z survey aimed to characterize the cool, photoionized gas at 2.2 ≲ z ≲ 3.6 in 202 H i -selected absorbers with 14.6 ≤ log N H I < 20 that probe the interface between galaxies and the intergalactic medium (IGM). We find that gas with 14.6 ≤ log N H I < 20 at 2.2 ≲ z ≲ 3.6 can be metal-rich (âˆ'1.6 ≲ [X/H] ≲ âˆ' 0.2) as seen in damped Ly α absorbers (DLAs); it can also be very metal-poor ([X/H] < âˆ' 2.4) or even pristine ([X/H] < âˆ' 3.8), which is not observed in DLAs but is common in the IGM. For 16 < log N H I < 20 absorbers, the frequency of pristine absorbers is about 1%â€"10%, while for 14.6 ≤ log N H I ≤ 16 absorbers it is 10%â€"20%, similar to the diffuse IGM. Supersolar gas is extremely rare (<1%) at these redshifts. The factor of several thousand spread from the lowest to highest metallicities and large metallicity variations (a factor of a few to >100) between absorbers separated by less than Î" v < 500 km sâˆ'1 imply that the metals are poorly mixed in 14.6 ≤ log N H I < 20 gas. We show that these photoionized absorbers contribute to about 14% of the cosmic baryons and 45% of the cosmic metals at 2.2 ≲ z ≲ 3.6. We find that the mean metallicity increases with N H i , consistent with what is found in z < 1 gas. The metallicity of gas in this column density regime has increased by a factor ∼8 from 2.2 ≲ z ≲ 3.6 to z < 1, but the contribution of the 14.6 ≤ log N H I < 19 absorbers to the total metal budget of the universe at z < 1 is a quarter of that at 2.2 ≲ z ≲ 3.6. We show that FOGGIE cosmological zoom-in simulations have a similar evolution of [X/H] with N H i , which is not observed in lower-resolution simulations. In these simulations, very metal-poor absorbers with [X/H] < âˆ' 2.4 at z ∼ 2â€"3 are tracers of inflows, while higher-metallicity absorbers are a mixture of inflows and outflows. [ABSTRACT FROM AUTHOR]