The [CII]158um line is one of the dominant cooling lines in star-forming active regions. The commonly assumed clumpy UV-penetrated cloud models predict a [CII] line profile similar to that of CO. However, recent spectral-resolved observations show that they are often very different, indicating a more complex origin of the line emission including the dynamics of the source region. The aim of our study is to investigate the physical properties of the star-forming ISM in the Large Magellanic Cloud (LMC) by separating the origin of the emission lines spatially and spectrally. In this paper, we focus on the spectral characteristics and the origin of the emission lines, and the phases of carbon-bearing species in the N159 star-forming region in the LMC. We mapped a 4x(3-4) region in N159 in [CII]158um and [NII]205um with the GREAT on board SOFIA, and in CO(3-2), (4-3), (6-5), 13CO(3-2), and [CI]3P1-3P0 and 3P2-3P1 with APEX. The emission of all transitions observed shows a large variation in the line profiles across the map and between the different species. At most positions the [CII] emission line profile is substantially wider than that of CO and [CI]. We estimated the fraction of the [CII] integrated line emission that cannot be fitted by the CO line profile to be 20%-50%. We derived the relative contribution from C+, C, and CO to the column density in each velocity bin. The contribution from C+ dominates the velocity range far from the velocities traced by the dense molecular gas, and the region located between the CO cores of N159 W and E. We estimate the contribution of the ionized gas to the [CII] emission using the ratio to the [NII] emission to be < 19% to the [CII] emission at its peak position, and <15% over the whole observed region. Using the integrated line intensities, we present the spatial distribution of I([CII])/I(FIR). (abridged for arXiv)
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