Can developing twins further divide and subdivide and become identical quadruplets or octuplets or...? Can twinning be induced, either in utero or in vitro? Might it be possible to just keep harvesting a few cells off the original clump indefinitely and making an unending stream of identical embryos for implantation in a surrogate uterus?
Yes. Up to the limit of what the mother's body can tolerate. Of course, if the cells never aggregate, that suggests some sort of fundamental problem with the embryo's developmental process, and you will probably not get a viable organism.
Sort of and yes. In utero, fertility drugs can cause multiple eggs to be released per cycle, which can cause fraternal twins when each egg is fertilized. IVF can do the same thing, but you can also manually subdivide a single embryo into two (smaller) viable embryos, provided you perform the division before any cell differentiation occurs.
Yes, but good luck getting an ethics review board to approve that :D