The survey of the mid-infrared sky by the Wide-field Infrared Survey Explorer (WISE) led to the discovery of extremely cold, low-mass brown dwarfs, classified as Y dwarfs, which extend the T class to lower temperatures. Twenty-four Y dwarfs are known at the time of writing. Here we present improved parallaxes for four of these, determined using Spitzer images. We give new photometry for four late-type T and three Y dwarfs and new spectra of three Y dwarfs, obtained at Gemini Observatory. We also present previously unpublished photometry taken from HST, ESO, Spitzer, and WISE archives of 11 late-type T and 9 Y dwarfs. The near-infrared data are put onto the same photometric system, forming a homogeneous data set for the coolest brown dwarfs. We compare recent models to our photometric and spectroscopic data set. We confirm that nonequilibrium atmospheric chemistry is important for these objects. Nonequilibrium cloud-free models reproduce well the near-infrared spectra and mid-infrared photometry for the warmer Y dwarfs with 425<=T_eff_(K)<=450. A small amount of cloud cover may improve the model fits in the near-infrared for the Y dwarfs with 325<=T_eff_(K)<=375. Neither cloudy nor cloud-free models reproduce the near-infrared photometry for the T_eff_=250K Y dwarf WISEJ085510.83-071442.5 (W0855). We use the mid-infrared region, where most of the flux originates, to constrain our models of W0855. We find that W0855 likely has a mass of 1.5-8 Jupiter masses and an age of 0.3-6Gyr. The Y dwarfs with measured parallaxes are within 20pc of the Sun and have tangential velocities typical of the thin disk. The metallicities and ages we derive for the sample are generally solar-like. We estimate that the known Y dwarfs are 3 to 20 Jupiter-mass objects with ages of 0.6-8.5Gyr.