For matrices with iid columns, the expectations of the title are examined using ideas from convex geometry. In particular, a representation is shown which involves random realizations from the class of convex bodies called zonoids. This result is used to derive bounds and comparisons of various types.
"Expected Absolute Random Determinants and Zonoids." Ann. Appl. Probab. 1 (2) 293 - 300, May, 1991. https://doi.org/10.1214/aoap/1177005938