In the string theory landscape, the set of particles we observe, their masses and interaction strengths originate from one of many different possible compactifications. What fundamental physical reality/principles/laws is the same in all string theory compactifications/universes in the landscape?
One answer to your question, which laws of physics should be reproduced in any string compactification, could be the following: If the Weak Gravity Conjecture, see http://arxiv.org/abs/hep-th/0601001, holds true, then gravity should come out as the weakest force in any string compactification. (Notice that there are - as far as I know - only arguments for the WGC coming from generic considerations of a theory of Quantum Gravity.)
Moreover, one can hope to find constraints on certain SM or cosmological parameters from string compactifications. For instance, the energy scale of inflation must not be too high, b/c otherwise one risks destabilisation of the extra dimensions. This translates into upper bounds on the field range of the inflaton. (By the way: If the WGC is true, then certain scenarios of natural inflation cannot be realised.)
these fundamental properties are unexplained and the same in all compactifications: totalitarian principle. speed of light. space has quantum jitters. space can expand. space can create particles. particles experience up to 4 forces at low energy, 1 of them know about all particles, 3 of them know about only some particles