We present ALMA CII 158 $\mu$m line and underlying far-infrared (FIR) continuum emission observations ($0^{\prime \prime }.70\times 0^{\prime \prime }.56$ resolution) toward HSC J124353.93$+$010038.5 (J1243$+$0100) at $z=7.07$, the only low-luminosity ($M_{1450}>-25$ mag) quasar currently known at $z>7$. The FIR continuum is bright (1.52 mJy) and resolved with a total luminosity of $L_{\mathrm{FIR}}=3.5\times 10^{12}{L}_{\odot }$. The spatially extended component is responsible for $\sim 40\%$ of the emission. The area-integrated CII spectrum shows a broad wing ($\mathrm{FWHM}=997$ km s${}^{-1}$, $L_{[\mathrm{CII}]}=1.2\times 10^9{L}_{\odot }$) as well as a bright core ($\mathrm{FWHM}=235$ km s${}^{-1}$, $L_{[\mathrm{CII}]}=1.9\times 10^9{L}_{\odot }$). This wing is the first detection of a galactic-scale quasar-driven outflow (atomic outflow rate $>447M_{\odot }$ yr${}^{-1}$) at $z>7$. The estimated large mass loading factor of the total outflow (e.g., $\gtrsim 9$ relative to the CII-based SFR) suggests that this outflow will soon quench the star-formation of the host. The core gas dynamics are governed by rotation, with a rotation curve suggestive of a compact bulge ($\sim 3.3\times 10^{10}{M}_{\odot }$), although it is not yet spatially resolved. Finally, we found that J1243$+$0100 has a black hole mass-to-dynamical mass ratio (and -to-bulge mass ratio) of $\sim 0.4\%$ ($\sim 1\%$), consistent with the local value within uncertainties. Our results therefore suggest that the black hole-host co-evolution relation is already in place at $z\sim 7$ for this object.