We have used the Röentgensatellit (ROSAT), the Extreme Ultraviolet Explorer (EUVE), and the Hubble Space Telescope (HST) to measure X-ray and ultraviolet emissions of moderate-mass (~2-3 M☉) giants in the Hertzsprung gap (spectral types early F to mid-G) and the post-helium flash ``clump'' (~G8-K0). Our motivation was to document the evolution of hot coronae (T > 106 K) along the post-main-sequence trajectories traveled by such stars in order to gain insight concerning the ``X-ray deficiency'' of the F-G0 giants and the strong braking of stellar rotation at the red edge of the Hertzsprung gap.With few exceptions, Hertzsprung gap and clump giants observed by ROSAT PSPC show hot (T ~ 107 K) coronal energy distributions, regardless of any X-ray deficiency. EUVE spectra of gap star 31 Com (G0 III) indicate a broad coronal emission measure hump at ~107.2 K, while the active clump giant β Ceti (K0 III) displays a sharp peak at ~106.8 K, as seen previously in the mixed clump/gap binary Capella (α Aur: G8 III + G0 III). The gap giants υ Peg (F8 III) and 24 UMa (G4 III) have EUV emissions of intermediate temperature (~107.0 K). The stars 31 Com, ψ3 Psc (G0 III), and β Cet exhibit redshifted transition zone (TZ: ~105 K) lines in HST GHRS spectra, as reported earlier in Procyon (α CMi: F5 IV-V) and Capella G0. Such redshifts on the Sun are thought to signify flows in magnetic loops. β Cas (F2 III)--a rare soft coronal source among the gap stars--displays blueshifts of C IV and O IV], although emissions at cooler and hotter temperatures are near the photospheric velocity. The remarkably broad line profiles of the fastest rotating gap giants suggest that the 105 K ``subcoronal'' emission zones extend to h~R* above the photosphere, about 50 scale heights. In contrast to the TZ line redshifts, the upper chromospheric emissions (e.g., Mg II and Si III) of 31 Com and ψ3 Psc have blueshifted cores. Blue-asymmetric peaks in the solar Mg II lines are thought to indicate dynamical heating in the chromosphere. Observations of the H I Lyα feature of 31 Com taken 9 months apart reveal striking profile changes, reminiscent of those noted previously in the Lyα blue peak of the Capella G0 star. We used the far-ultraviolet diagnostics, in combination with ROSAT X-ray photometry and EUVE high-excitation line strengths, to constrain physical models of the stellar outer atmospheres. Quasi-static magnetic loops can simulate the empirical coronal emission measures of the giant stars, but the inferred pressures for sensible loop lengths conflict with direct measurements of subcoronal densities. Furthermore, the high rate of emission at ~105 K cannot be explained by thermal conduction down the legs of hot quasi-static loops. On the other hand, the possible existence of elongated (l ~ R*) emission structures on the gap giants leads to a speculative scenario to explain the X-ray deficiency. It is based on the increased importance of the dynamical filling phase (``explosive evaporation'') of the loop life cycle; conductive cooling, yielding TZ emissions at the footpoints, when the heating is interrupted; and the possibility for transitions between ``hot'' and ``cool'' energy balance solutions owing to dynamical suspension and centrifugal trapping of the cooling gas. The long loops might represent a vestigial global ``magnetosphere'' inherited from the main-sequence phase, which ultimately is disrupted near ~G0 by the deepening convective envelope and growth of a more solar-like dynamo. Coronal emissions might be boosted temporarily as the X-ray deficiency is removed but soon would be quenched by wind braking previously inhibited by the magnetospheric ``dead zone.''