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We present detailed and comprehensive data reduction andpoint-spread-function (PSF) model construction for all public JWST NIRCamimaging data from the COSMOS-Web treasury program (up to June 2023, totaling0.28 ${\rm deg}^2$). We show that the NIRCam PSF has significantshort-timescale temporal variations and random spatial variations in all fourfilters (F115W, F150W, F277W, and F444W). Combining NIRCam with archival HSTimaging, we perform multiwavelength AGN+host image decomposition to study theproperties of 143 X-ray-selected ($L_{\rm bol}=10^{43.6-47.2}$ erg s$^{-1}$)broad-line AGNs at $0.35\lesssim z \lesssim 3.5$. Leveraging the superbresolution, wavelength coverage, and sensitivity of NIRCam, we successfullydetect host stellar emission after decomposing the central AGN point source in142 objects. $\sim 2/3$ AGNs are in star-forming galaxies based on the UVJdiagram, suggesting no instantaneous negative AGN feedback. X-ray-selectedbroad-line AGN hosts follow a similar stellar mass-size relation as inactivegalaxies, albeit with slightly smaller galaxy sizes. We find that althoughmajor mergers are rare ($\sim$7-22%) among the sample, more subtlenon-axisymmetric features from stellar bars, spiral arms, and minor mergers areubiquitous, highlighting the importance of secular processes and minor mergersin triggering AGN activity. For a subsample of 30 AGNs at $1<z<2.5$ with blackhole mass measurements from single epoch spectra, they follow a similar blackhole mass-stellar mass relation as local inactive early-type galaxies butreside preferentially near the upper envelope of nearby AGNs. We caution thatselection biases and intrinsic differences of AGN populations at differentredshifts may significantly affect their location on the black holemass-stellar mass plane.

AGNs and Host Galaxies in COSMOS-Web. I. NIRCam Images, PSF Models and  Initial Results on X-ray-selected Broad-line AGNs at $0.35\lesssim z \lesssim  3.5$