We derive basic analytical results for the timing and decay of the GRB-counterpart and delayed-afterglow light-curves for a brief emission episode from a relativistic surface endowed with angular structure, consisting of a uniform Core of size theta_c (Lorentz factor Gamma_c and surface emissivity i_nu are angle-independent) and an axially-symmetric power-law Envelope (Gamma ~ theta^{-g}). In this Large-Angle Emission (LAE) model, radiation produced during the prompt emission phase (GRB) at angles theta > theta_c arrives at observer well after the burst (delayed emission). The dynamical time-range of the very fast-decaying GRB tail and of the flat afterglow plateau, and the morphology of GRB counterpart/afterglow, are all determined by two parameters: the Cores parameter Gamma_c*theta_c and the Envelopes Lorentz factor index g, leading to three types of light-curves that display three post-GRB phases (type 1: tail, plateau/slow-decay, post-plateau/normal-decay), two post-GRB phases (type 2: tail and fast-decay), or just one (type 3: normal decay). We show how X-ray light-curve features can be used to determine Core and Envelope dynamical and spectral parameters. Testing of the LAE model is done using the Swift/XRT X-ray emission of two afterglows of type 1 (060607A, 061121), one of type 2 (061110A), and one of type 3 (061007). We find that the X-ray afterglows with plateaus require an Envelope Lorentz factor Gamma ~ theta^{-2} and a comoving-frame emissivity i_nu ~ theta^2, thus, for a typical afterglow spectrum F_nu ~ nu^{-1}, the lab-frame energy release is uniform over the emitting surface.
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