Air-based molecular communication (MC) has the potential to be one of the first MC systems to be deployed in real-world applications, enabled by commercially available sensors. However, these sensors usually exhibit non-linear and cross-reactive behavior, contrary to the idealizing assumption of linear and perfectly molecule type-specific sensing often made in the MC literature. To address this mismatch, we propose several detectors and transmission schemes for a molecule mixture communication system where the receiver (RX) employs non-linear, cross-reactive sensors. All proposed schemes are based on the first- and second-order moments of the symbol likelihoods that are fed through the non-linear RX using the Unscented Transform. In particular, we propose an approximate maximum likelihood (AML) symbol-by-symbol detector for inter-symbol-interference (ISI)-free transmission scenarios and a complementary mixture alphabet design algorithm which accounts for the RX characteristics. When significant ISI is present at high data rates, the AML detector can be adapted to exploit statistical ISI knowledge. Additionally, we propose a sequence detector which combines information from multiple symbol intervals. For settings where sequence detection is not possible due to extremely limited computational power at the RX, we propose an adaptive transmission scheme which can be combined with symbol-by-symbol detection. Using computer simulations, we validate all proposed detectors and algorithms based on the responses of commercially available sensors as well as artificially generated sensor data incorporating the characteristics of metal-oxide semiconductor sensors. By employing a general system model that accounts for transmitter noise, ISI, and general non-linear, cross-reactive RX arrays, this work enables reliable communication for a large class of MC systems.