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We explore several aspects of the current fluctuations and correlations in the box-ball system (BBS), an integrable cellular automaton in one space dimension. The state we consider is an ensemble of microscopic configurations where the box occupancies are independent random variables (i.i.d. state), with a given mean ball density. We compute several quantities exactly in such homogeneous stationary state: the mean value and the variance of the number of balls $N_t$ crossing the origin during time $t$, and the scaled cumulants generating function associated to $N_t$. We also compute two spatially integrated current-current correlations. The first one, involving the long-time limit of the current-current correlations, is the so-called Drude weight and is obtained with thermodynamic Bethe Ansatz (TBA). The second one, involving equal time current-current correlations is calculated using a transfer matrix approach. A family of generalized currents, associated to the conserved charges and to the different time evolutions of the models are constructed. The long-time limits of their correlations generalize the Drude weight and the second cumulant of $N_t$ and are found to obey nontrivial symmetry relations. They are computed using TBA and the results are found to be in good agreement with microscopic simulations of the model. TBA is also used to compute explicitly the whole family of flux Jacobian matrices. Finally, some of these results are extended to a (non-i.i.d.) two-temperatures generalized Gibbs state (with one parameter coupled to the total number of balls, and another one coupled to the total number of solitons).