Power-to-methane technologies have been regarded as a promising alternative to offer small- or large-scale, long-timescale (daily/weekly/seasonal) energy storage as well as the opportunity of utilizing CO2. The performance of the core component, the electrolyzer, largely determines how well a power-to-methane system can perform, making high-temperature solid-oxide electrolysis attractive because of its inherent high electrical efficiency. More importantly, solid-oxide electrolysis uniquely allows co-electrolysis of steam and CO2 for producing syngas, the composition of which can be readily, flexibly adjusted to synthesize different hydrocarbon fuels. In this paper, for both steam- and co-electrolysis, we comprehensively and comparatively investigate several critical design issues of a solid-oxide electrolyzer based power-to-methane system with fixed-bed methanation reactor and membrane-based methane upgrading: (1) system-level heat integration, (2) the impacts of operating variables (e.g., operating voltage, reactant utilization, anode/cathode feed ratio, and operating pressure of the methanation reactor and membrane) on system performances, (3) the competitiveness of the electrolyzer operation with pure oxygen production, and (4) the possibility of avoiding electrical heating, which is necessary for thermoneutral operation to heat up the electrolyze" feeds to the required temperature. To achieve this target, a multi-objective optimization platform with integrated heat cascade calculation is employed with experimentally-calibrated component models. The results show that, for both steam- and co-electrolysis, there is a trade-off between system efficiency and methane yield: pursuing a higher efficiency generally reduces the methane yield, which is a consequence of electrochemistry, stack cooling and system-level heat integration. Instead of sweep air, pure oxygen production is preferred only at small current density, which delivers the highest system efficiency but the lowest methane yield. When the electrolyzer operates exothermically, methane production and the total power consumption can vary in much wider ranges than those with the electrolyzer operating under thermoneutral mode, which leads to potential enhancement of operation flexibility and reliability. The co-electrolysis coupling with strongly-exothermic syngas methanation, in general, offers better heat integration opportunity with sweep air, but less with pure oxygen production. In addition, several design heuristics, e.g., the operating pressure of the electrolyzer and methanation reactor, are concluded to potentially guide practical applications.