What differences will I get if I shoot the scene with a narrow-angle lens (from the same position as I would have done with the wide lens) and stitch the images with Hugin or Autostitch, e.g. in terms of relative proportions of near and far objects and depth of field?
The differences are those dictated by the lens. A stitched panorama typically won't have the "feel" of a wide angle lens, because it won't have the barrel distortion such a lens typically exhibits, and the DoF will be determined by the lens/aperture setting/subject distance you're using. For landscape usage, in the f/8-f/16 arena, chances are that depth of field won't be noticeably different, but stitching will sometimes create more background blur with longer lenses at wider apertures, which is why portrait photographers sometimes go Brenizer Method.
For my tastes, ultrawide and wide angle lenses tend to exhibit more "funk", while a stitched panorama (assuming a not super-wide typical landscape pano) exhibits greater "calm." One isn't necessarily better than the other but they do taste a little different.
And, of course, there's the issue of ghosts/clones with moving subjects with any post-processing method that involves combining multiple images.
What stitching mode (e.g. spherical, cylindrical or rectilinear) should I use to best simulate a wide-angle lens?
It depends on how much distortion you like, what kind of distortion you like, and the angle of view of the final panorama. Cylindrical is most likely to be your go-to if you're shooting not-super-wide landscape panoramas that are a handful of member images. Equirectangular/spherical works better if you're doing 360x180s spherical panos, but stereographic is a good place to go if you're going so wide that cylindrical and rectilinear are causing issues. Rectilinear, oddly, may be the worst choice of all for very wide panoramas, because of the extreme distortion and shape that will result (the image gets pulled into an X-like shape), and is probably only going to be good if you're shooting a small number of images with a normalish lens or using a telephoto lens. However, none of these projections is particularly good at simulating an ultrawide, because in most stitching programs lens distortion is corrected for prior to stitching, and then mapped out along one of these projections, none of which really simulate an ultrawide lens with barrel distortion. Fisheye can simulate a fisheye's equisolid mapping pretty well, but that's far more extreme than an ultrawide lens will give you.
If what you really really really want is the effect of shooting with an ultrawide lens, I'd say go get an ultrawide lens, and don't bother with panorama stitching. OTOH, you may find that panorama stitching is its own reward, rather than a mere ultrawide substitute.