BlackHoleWeather aims to unify the astrophysics of black-hole (BH) feeding and feedback within cosmic structures, in one comprehensive theory that leverages novel high-performance simulations, fundamental gas physics, and timely multiwavelength observations. Most of the ordinary matter in the Universe is in the form of a tenuous gas which fills galaxies, groups, and clusters of galaxies (circumgalactic, intragroup, and intracluster medium). Such cosmic atmospheres are shaped by complex thermo-hydrodynamical processes - akin to Earth weather - with the central BH acting as cosmic thermostat over scales of 9 orders of magnitude. We have entered a Golden Age of multiphase gas detections continuously discovering ionized filaments (optical/UV), neutral gas (IR/21cm), and molecular clouds (radio) which condense out of the hot X-ray halos or that are ejected via BH feedback. We will tackle key challenges of modern astrophysics: what is the origin and evolution of the macro precipitation; how the multiphase rain (or chaotic cold accretion) is fed down through the BH horizon; how matter/energy is re-ejected back by the BH and deposited via multiphase outflows, jets and radiation; what is the role of dust, turbulence, stars, and cosmic rays; and how the self-regulated BH feeding-feedback loop shapes galaxies throughout cosmic time. Bridging BH feeding and feedback via ab-initio, multi-scale (mpc to Mpc), and first-principle physics (magnetohydrodynamics, transport, chemistry, cosmology) is ambitious, yet it is a zero-to-one leap that current astrophysics must undertake, and whose public datasets will provide invaluable legacy for many astronomical communities. BlackHoleWeather is a frontier yet feasible project, exploiting the timely convergence of our groundbreaking massively-parallel GPU code (GAMER2) and our ongoing multifrequency observing programs (e.g., Chandra, XMM, HST, ALMA, MUSE, JWST, SOFIA, MeerKAT).