Searches for scalar leptoquarks pair-produced in proton-proton collisions at $s=13$ TeV at the Large Hadron Collider are performed by the ATLAS experiment. A data set corresponding to an integrated luminosity of 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ is used. Final states containing two electrons or two muons and two or more jets are studied, as are states with one electron or muon, missing transverse momentum and two or more jets. No statistically significant excess above the Standard Model expectation is observed. The observed and expected lower limits on the leptoquark mass at 95% confidence level extend up to 1.29 TeV and 1.23 TeV for first- and second-generation leptoquarks, respectively, as postulated in the minimal Buchm"uller-R"uckl-Wyler model, assuming a branching ratio into a charged lepton and a quark of 50%. In addition, measurements of particle-level fiducial and differential cross sections are presented for the $Z\to ee$, $Z\to \mu \mu$ and $tt¯$ processes in several regions related to the search control regions. Predictions from a range of generators are compared with the measurements, and good agreement is seen for many of the observables. However, the predictions for the $Z\to \ell \ell$ measurements in observables sensitive to jet energies disagree with the data.