We present an in-depth experimental-computational study of the parameters necessary to optimize a tunable, quasi-monoenergetic, efficient, low-background Compton backscattering (CBS) x-ray source that is based on the self-aligned combination of a laser-plasma accelerator (LPA) and a plasma mirror (PM). The main findings are: (1) an LPA driven in the blowout regime by 30 TW, 30 fs laser pulses producesnot only a high-quality, tunable, quasi-monoenergetic electron beam, but also a high-quality, relativistically intense (a0~1) spent drive pulse that remains stable in profile and intensity over the LPA tuning range. (2) A thin plastic film near the gas jet exit retro-reflects the spent drive pulse efficiently into oncoming electrons to produce CBS x-rays without detectable bremsstrahlung background. Meanwhile anomalous far-field divergence of the retro-reflected light demonstrates relativistic "denting" of the PM. Exploiting these optimized LPA and PM conditions, we demonstrate quasi-monoenergetic (50% FWHM energy spread), tunable (75 to 200 KeV) CBS x-rays, characteristics previously achieved only on more powerful laser systems by CBS of a split-off, counter-propagating pulse. Moreover, laser-to-x-ray photon conversion efficiency ~6e12 exceeds that of any previous LPA-based quasi-monoenergetic Compton source. Particle-in-cell simulations agree well with the measurements. ; Comment: 10 pages, 7 figures