Yes, but similar to the classification of regular solids there are few such polytopes as the dimension gets high enough. There are 5 four dimensional deltatopes and only 3 for each higher dimension (the simplex, the cross-polytope, and the bipyramid over the lower dimensional simplex). This is proved in Sullivan's (unpublished) preprint "Convex Deltatopes in all Dimensions and Polyhedral Soap Films" (available here).

Yes, but similar to the classification of regular solids there are few such polytopes as the dimension gets high enough. There are 5 four dimensional deltatopes and only 3 for each higher dimension (the simplex, the cross-polytope, and the bipyramid over the lower dimensional simplex). This is proved in Sullivan's (unpublished) preprint "Convex Deltatopes in all Dimensions and Polyhedral Soap Films" (available here).