(Abridged) The 30 Dor region in the Large Magellanic Cloud (LMC) is the most vigorous star-forming region in the Local Group. Star formation in this region is taking place in low-metallicity molecular gas that is exposed to an extreme far--ultraviolet (FUV) radiation field powered by the massive compact star cluster R136. We used the NANTEN2 telescope to obtain high-angular resolution observations of the 12CO 4-3, 7-6, and 13CO 4-3 rotational lines and [CI] 3P1-3P0 and 3P2-3P1 fine-structure submillimeter transitions in 30Dor-10, the brightest CO and FIR-emitting cloud at the center of the 30Dor region. We derived the properties of the low-metallicity molecular gas using an excitation/radiative transfer code and found a self-consistent solution of the chemistry and thermal balance of the gas in the framework of a clumpy cloud PDR model. We compared the derived properties with those in the N159W region, which is exposed to a more moderate far-ultraviolet radiation field compared with 30Dor-10, but has similar metallicity. We also combined our CO detections with previously observed low-J CO transitions to derive the CO spectral-line energy distribution in 30Dor-10 and N159W. The separate excitation analysis of the submm CO lines and the neutral carbon fine structure lines shows that the mid-J CO and [CI]-emitting gas in the 30Dor-10 region has a temperature of about 160 K and a H2 density of about 10^4 cm^-3. We find that the molecular gas in 30Dor-10 is warmer and has a lower beam filling factor compared to that of N159W, which might be a result of the effect of a strong FUV radiation field heating and disrupting the low--metallicity molecular gas. We use a clumpy PDR model (including the [CII] line intensity reported in the literature) to constrain the FUV intensity to about chi_0 ~ 3100 and an average total H density of the clump ensemble of about 10^5 cm^-3 in 30Dor-10.
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