This dissertation investigates Stanley-Reisner rings and monomial ideals in connection to some important concepts in characteristic p commutative algebra, such as Frobenius complexity, and complexity sequence, and strong test ideals in tight closure theory. The Frobenius complexity of a local ring R measures asymptotically the abundance of Frobenius operators of order e on the injective hull of the residue field of R. It is known that, for Stanley-Reisner rings, the Frobenius complexity is either negative infinity or zero. This invariant is determined by the complexity sequence of the ring of Frobenius operators on the injective hull of the residue field. One of our main results shows that the complexity sequence is constant starting with its second term, generalizing work of Alvarez Montaner, Boix and Zarzuela. This result settles an open question mentioned by Alvarez Montaner in one of his papers. Moreover, we use Cartier algebras to describe a large class of strong test ideals. One of our main results gives a full description of test ideals associated to Cartier algebras in Stanley-Reisner rings. An important consequence of our result states that a bound for the degree of integral dependence that an arbitrary element in the tight closure of an ideal satisfies over the respective ideal is given by a combinatorial invariant, which is the number of facets of the Stanley-Reisner ring considered.