Kidney exchanges allow patients with end-stage renal disease to find a lifesaving living donor by way of an organized market. However, not all patients are equally easy to match, nor are all donor organs of equal quality---some patients are matched within weeks, while others may wait for years with no match offers at all. We propose the first decision-support tool for kidney exchange that takes as input the biological features of a patient-donor pair, and returns (i) the probability of being matched prior to expiry, and (conditioned on a match outcome), (ii) the waiting time for and (iii) the organ quality of the matched transplant. This information may be used to inform medical and insurance decisions. We predict all quantities (i, ii, iii) exclusively from match records that are readily available in any kidney exchange using a quantile random forest approach. To evaluate our approach, we developed two state-of-the-art realistic simulators based on data from the United Network for Organ Sharing that sample from the training and test distribution for these learning tasks---in our application these distributions are distinct. We analyze distributional shift through a theoretical lens, and show that the two distributions converge as the kidney exchange nears steady-state. We then show that our approach produces clinically-promising estimates using simulated data. Finally, we show how our approach, in conjunction with tools from the model explainability literature, can be used to calibrate and detect bias in matching policies.