Aims: In this paper, we perform detailed modelling of the Spitzer and Herschel observations of the LMC, in order to: (i) systematically study the uncertainties and biases affecting dust mass estimates; and to (ii) explore the peculiar ISM properties
of the LMC. Methods: To achieve these goals, we have modelled the spatially resolved SEDs with two alternate grain compositions, to study the impact of different submillimetre opacities on the dust mass. We have rigorously propagated the observational errors (noise and calibration) through the entire fitting process, in order to derive consistent parameter uncertainties. Results: First, we show that using the integrated SED leads to underestimating the dust mass by ~50 % compared to the value obtained with sufficient spatial resolution, for the region we studied. This might be the case, in general, for unresolved galaxies. Second, we show that Milky Way type grains produce higher gas-to-dust mass ratios than what seems possible according to the element abundances in the LMC. A spatial analysis shows that this dilemma is the result of an exceptional property: the grains of the LMC have on average a larger intrinsic submm opacity (emissivity index beta~1.7 and opacity kappa_abs(160 microns)=1.6 m2/kg) than those of the Galaxy. By studying the spatial distribution of the gas-to-dust mass ratio, we are able to constrain the fraction of unseen gas mass between ~10, and ~100 % and show that it is not sufficient to explain the gas-to-dust mass ratio obtained with Milky Way type grains. Finally, we confirm the detection of a 500 microns extended emission excess with an average relative amplitude of ~15 %, varying up to 40 %. This excess anticorrelates well with the dust mass surface density. Although we do not know the origin of this excess, we show that it is unlikely the result of very cold dust, or CMB fluctuations.
