What effect do different SMT stencil materials have on results?
A stainless steel stencil will help, probably a good deal, with the printing of fine pitch parts.  The ones with the bends on the sides (like the ones you pointed to) tend to be stiffer than the ones without the bends (like the ones that pcb-pool include for free with board orders.  The latter can bow in the center and cause some problems with printing.  I don't think it will help as much with production time as you want it to, unless you count wiping off the bad prints and redoing it with the plastic stencils.

Depending on your level of production, you might consider a real manual stencil printer for use with framed stencils, like 

These, of course, use framed stencils, so you'll have a $500 one time cost for the printer, and about $300 NRE for the framed stencil.  Storage of the framed stencils may eventually become an issue.

There are also steps you can take in board design that can help.  Make sure there's room for solder mask in between pins on fine pitch parts, if you can, and your process will be a bit more forgiving.  Also, adjust the apertures in the cream layer to be a bit smaller if you're hand printing with less than great stencils.  This can be a pain to do, especially if you want to do it only for SOME parts using some PCB packages.  In eagle, for example, for fine pitch parts I set SMD pads as NOCREAM, and I add the cream aperture manually for each pad (I generate a macro to do this in octave, and run the macro in eagle).