In this note, we investigate some of the fundamental algebraic and geometric properties of s-lecture hall simplices and their generalizations. We show that all s-lecture hall order polytopes, which simultaneously generalize s-lecture hall simplices and order polytopes, satisfy a property which implies the integer decomposition property. This answers one conjecture of Hibi, Olsen and Tsuchiya. By relating s-lecture hall polytopes to alcoved polytopes, we then use this property to show that families of s-lecture hall simplices admit a quadratic Gröbner basis with a square-free initial ideal. Consequently, we find that all s-lecture hall simplices for which the first order difference sequence of s is a 0, 1-sequence have a regular and unimodular triangulation. This answers a second conjecture of Hibi, Olsen and Tsuchiya, and it gives a partial answer to a conjecture of Beck, Braun, Köppe, Savage and Zafeirakopoulos.