Mixed, charge and heat current fluctuations as well as thermoelectric differential conductancesare considered for non-interacting nanosystems connected to reservoirs. Using theLandauer-Buttiker formalism, we derive general expressions for these quantities and considertheir possible relationships in the entire ranges of temperature, voltage and coupling to theenvironment or reservoirs. We introduce a dimensionless quantity given by the ratio betweenthe product of mixed noises and the product of charge and heat noises, distinguishing betweenthe auto-ratio defined in the same reservoir and the cross-ratio between distinct reservoirs.From the linear response regime to the high-voltage regime, we further specify the analyticalexpressions of differential conductances, noises and ratios of noises, and examine theirbehavior in two concrete nanosystems: a quantum point contact in an ohmic environment anda single energy level quantum dot connected to reservoirs. In the linear response regime, wefind that these ratios are equal to each other and are simply related to the figure of merit. Theycan be expressed in terms of differential conductances with the help of thefluctuation-dissipation theorem. In the non-linear regime, these ratios radically distinguishbetween themselves as the auto-ratio remains bounded by one, while the cross-ratio exhibitsrich and complex behaviors. In the quantum dot nanosystem, we moreover demonstrate thatthe thermoelectric efficiency can be expressed as a ratio of noises in the non-linear Schottkyregime. In the intermediate voltage regime, the cross-ratio changes sign and diverges, whichevidences a change of sign in the heat cross-noise.