Context : Herschel  provides crucial constraints on the IR SEDs of galaxies , allowing unprecedented accuracy on the dust mass estimates . However , these estimates rely on non-linear models and poorly-known optical properties . 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 \simeq 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 \simeq 1.7 and opacity \kappa _ { \mbox { { \scriptsize abs } } } ( 160 \mu { m } ) = 1.6 m ^ { 2 } kg ^ { -1 } ) 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 \simeq 10 , and \simeq 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 \mu { m }  extended emission excess with an average relative amplitude of \simeq 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 . Conclusions :