We present CO(2−1) and adjacent continuum observations of 7 nearby radio-quiet type-2 quasars (QSO2s) obtained with ALMA at ∼0.2′′ resolution (370 pc at z∼0.1). These QSO2s are luminous (L[OIII]>10^8.5 Lsun ~ MB <-23), and their host galaxies massive (M∗∼10^11 Msun). The CO morphologies are diverse, including discs with CO radii of rCO ∼4-5 kpc, interacting, merging and post-merger systems with rCO∼1.3–5 kpc, and red early-type galaxies (ETGs) with no CO(2−1) detected. In the interacting, merging and post-merger galaxies, the central kpc contains 18–25% of the total cold molecular gas, whereas in the spiral galaxies it is only ∼5–12%. J1010+0612 and J1430+1339 show double-peaked CO flux maps along the major axis of the CO discs, which do not have an optical counterpart at the same angular resolution. Based on our analysis of the ionized and molecular kinematics and mm continuum emission, these CO morphologies could be produced by AGN feedback in the form of outflows and/or jets/shocks. The CO kinematics of the QSO2s with CO(2−1) detections are dominated by rotation but also reveal non-circular motions. According to our analysis of the kinematics, these non-circular motions correspond to molecular outflows mostly coplanar with the CO discs in four of the QSO2s, and either to a coplanar inflow or vertical outflow in the case of J1010+0612. These molecular outflows represent 0.2–0.7% of the QSO2s total molecular gas mass, have maximum velocities of 200–350 km/s, radii from 0.4 to 1.3 kpc and outflow mass rates of 8–16 M/yr. These outflow properties are intermediate between those of the mild molecular outflows measured for Seyfert galaxies, and the fast and energetic outflows shown by ultra-luminous infrared galaxies. This suggests that it is not only AGN luminosity what drives massive molecular outflows. Other factors such as jet power, coupling between wind/jet/ionized outflows and CO discs, and amount/geometry of dense gas in the nuclear regions might be also relevant. Thus, although we do not find evidence for a significant impact of the QSO2s molecular outflows/AGN feedback on the total molecular gas reservoirs and SFRs, they would be modifying the distribution of cold molecular gas in the central kpc of the galaxies.