For PGD, and PGD-comparable removal for stem cell research, normally no more than 8 cells (3 days post-fertilization). Most cells taken for embryonic stem cell research have been taken from blastocysts, which are a little further along -- apparently they're easier to keep going, but it looks like this group has figured out how to get comparable results from earlier cells.
I don't think there's any reason that a blastocyst couldn't survive cell removal just as well as these earlier embryos, but it hasn't been the practice to try to save blastocysts when they're being used to provide cells for research. If PGD is done for the purpose of selecting embryos to transfer into a patient, it has to be done by the 8 cell/3 day stage because results aren't immediately available, and doing the test later would mean the embryos were too far along by the time they could be transferred, to have a good chance of surviving and implanting.
I don't think there's any reason that a blastocyst couldn't survive cell removal just as well as these earlier embryos
The issue isn't blastocyst survival but subsequent developmental progrsm being effected. I'd think there is a good reason to think this may be effected, but both of you have provided info that empirically it doesn't seem to.
Interestingly, it may be that later manipulation would be more deleterious than at this very early pre-implantation stage before any obvious differentiation and before pattern formation begins.