We present a comprehensive study of disks around 81 young low-mass stars and brown dwarfs in the nearby ~2-Myr-old Chamaeleon I star-forming region. We use mid-infrared photometry from the Spitzer Space Telescope, supplemented by findings from ground-based high-resolution optical spectroscopy and adaptive optics imaging. We derive disk fractions of 52 (+/-6) % and 58 (+6/-7) % based on 8-micron and 24-micron colour excesses, respectively, consistent with those reported for other clusters of similar age. Within the uncertainties, the disk frequency in our sample of K3-M8 objects in Cha I does not depend on stellar mass. Diskless and disk-bearing objects have similar spatial distributions. There are no obvious transition disks in our sample, implying a rapid timescale for the inner disk clearing process; however, we find two objects with weak excess at 3-8 microns and substantial excess at 24 microns, which may indicate grain growth and dust settling in the inner disk. For a sub-sample of 35 objects with high-resolution spectra, we investigate the connection between accretion signatures and dusty disks: in the vast majority of cases (29/35) the two are well correlated, suggesting that, on average, the timescale for gas dissipation is similar to that for clearing the inner dust disk. The exceptions are six objects for which dust disks appear to persist even though accretion has ceased or dropped below measurable levels. Adaptive optics images of 65 of our targets reveal that 17 have companions at (projected) separations of 10-80 AU. Of the five <20 AU binaries, four lack infrared excess, possibly indicating that a close companion leads to faster disk dispersal. The closest binary with excess is separated by ~20 AU, which sets an upper limit of ~8 AU for the outer disk radius. (abridged)
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