Frontier Fields : Combining HST, VLT and Spitzer data to explore the $z$$\sim$8 Universe behind the lensing cluster MACS0416$-$2403

(Abridged) The HST Frontier Fields project started at the end of 2013 with the aim of providing extremely deep images of 6 massive galaxy clusters. One of the main goals of this program is to push several telescopes to their limits in order to provide the best current view of the earliest stages of the Universe. We present a detailed analysis of $z$$\sim$8 objects behind the HFFs lensing cluster, MACS0416-2403, combining 0.3-1.6 $\mu$m imaging from HST, ground-based $K_s$ imaging from VLT HAWK-I, and 3.6 $\mu$m and 4.5 $\mu$m Spitzer Space Telescope. We apply the classical Lyman Break technique, combining non-detection criteria and color-selection. In order to avoid contamination by mid-$z$ interlopers, we require a strong break between optical and near infrared data. The luminosity function at $z$$\sim$8 is computed using a MC method taking advantage of the SED-fitting results. A piece of cautionary information is gleaned from new deep optical photometry of a previously identified $z$$\sim$8 galaxy in this cluster, which is now firmly detected as a mid-$z$ interloper with a strong $\approx1.5$ mag Balmer break (between F606W and F125W). Using the SED of this interloper, we estimate the contamination rate of our MACS0416$-$2403 sample, as well as for previous samples in Abell 2744 based on HFF data. Our selection recovers 4 robust objects with m$_{F160W}$ ranging from 26.0 to 27.9 AB and located in modest amplification regions ($\mu<$2.4). Two of the objects display a secondary break between the IRAC 3.6 $\mu$m and 4.5 $\mu$m bands which could be associated to the Balmer break or emission lines at $z$$\sim$8. The candidates generally have star formation rates around $\sim$10 M$_{\odot}$/yr and sizes ranging from 0.2 to 0.5 kpc, in good agreement with previous observations and expectations for objects in the early Universe.