It looks like the through holes on their QFP adapters are on a staggered 0.050" grid but there are no dimensions on their drawings. Is that true?
Yes - those holes accept parts like Arduino female headers.
Also, there are some rectangular pads near the chip area that appear to be for installing 0.050 pitch surface mount connectors or headers. Any ideas about that?
Hmm - maybe surface mount headers? Probe points?http://www.schmartboard.com/schmartboard_pd_202-0011-01.pdf
Yes, no dimension. Adjacent holes are 0.1" apart.
5 sets of pads in each corner too for surface mount to thruhole, maybe install parts there & jumper over to pins?
Looks like outer two rows will allow two 2x6 headers (2x12 total) to be used without having the holes be offset by 0.05", or a 2x13 with the left-end set of holes being the same signal