We present a theoretical study of electrical current fluctuations and finite-frequency noise in a double quantum dot connected to two electron reservoirs with the aim of deriving the Fano factor, the $\Delta T$-noise and the cross-correlations. This allows one to highlight several interesting features. Firstly the possibility of getting a significant reduction of current noise and Fano factor either when the system is placed in a given operating regime, or when a temperature gradient is applied between the two reservoirs, resulting from the fact that a negative $\Delta T$-noise is generated. The second feature is the sign change found in the cross-correlator between the two reservoirs with increasing frequencies. This study clarifies the understanding of the results obtained experimentally in such systems.