We study whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of D ≥ 3 CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, b, obeys a c-theorem, as in 2D CFT. For a 2D defect, we show that another, d2, interpreted as the defect's "conformal dimension," must be non-negative if the Averaged Null Energy Condition (ANEC) holds in the presence of the defect. We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by b and d2. Using this and known holographic results, we compute b and d2 for 1/2-BPS surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with b's c-theorem. Via free field and holographic examples we show that no universal "Cardy formula" relates the central charges to thermal entropy.