Intersection of hypercube and hyperplane - features of resulting polytope?

The nicest description I know of these polytopes is as intersections of a positive and a negative homothet of the regular simplex (having the same centre). The easiest way to see this is to think of the cube as the intersection of two orthants, namely the usual positive orthant and its reflection in the point $(\frac12,\dotsc,\frac12)$; the intersection of the cube with a hyperplane is the intersection of the respective intersections of these orthants with that hyperplane. The orthants are cones whose cross-sections are regular simplices of the next lower dimension. (For example, the convex hull of the standard basis vectors is a regular simplex of the next lower dimension.)

To expand on achille hui's comment using this mental model: It's a simplex near the corners because when you cut close enough to the corner, one of the homothets is small enough to fit entirely inside the other, so the cross-section is just the smaller homothet.

The 2-faces are not in general centrally symmetric: a suitable cross-section will give a simplex intersected with a large negative copy which is almost but not quite large enough to contain the first simplex, so it cuts off the vertices slightly. All the 2-faces touch some vertex, so they'll be irregular hexagons — again, as noted in achille hui's comment for the case $n=3$; you can also visualize the tetrahedron with its tips cut off to get the case $n=4$.