What is the window during which division into twins is possible (for instance can it still happen after implantation in the uterus)? Is one twin the "original" and the other its clone?
In the case of two healthy, viable identical twins, the separation generally occurs at the first division from one zygote to two cells. So they are equal partners in the split, not an original and a clone. During the first several cell divisions in the morula and blastula, there is no significant growth. So these divisions produce smaller and smaller cells. If one of these smaller cells is separated, it is very unlikely to be substantial enough to survive and implant in the uterus.
This page describes what happens pretty well:
Division into twins can apparently happen at several stages up to and including implantation, but not much after that. The later the division the worse the prognosis because the more the twins share. The article mentions three stages in particular: division at the 2-blastomere stage (i.e. when the original zygote is cleaving), which leads to two completely separate embryos that implant separately, division at the embryoblast stage which leads to the embryos sharing a chorionic cavity (i.e. placenta), and division at the embryonic disk stage, which happens after implantation and involves the fetuses sharing everything - apparently to the point that they rarely survive until birth because they get tangled in each other's umbilical cords. Later divisions, after the primitive streak starts forming (i.e. about two weeks after fertilization), can lead to conjoined twins, because the division happened when the cells have already started to differentiate into which part of the body they'll be.
In no case can one twin be considered "the original" and the other not. Twinning involves whatever stage we're at dividing into two; we can't talk about one being an offshoot of the other, anymore than if you pour cake batter into two molds instead of one you don't end up with an original cake and a copy, you just end up with two cakes. Whose batter happened to be in the same bowl until some point in the baking process.
Moreover the twinning events usually involve a symmetrical division, meaning we can't even talk of one twin having a claim to a higher percentage of the original embryonic material or whatever. An exception may be parasitic twins and other examples of asymmetrical twins, but I haven't been able to find a good source on what causes those - it's unclear whether it involves twins resulting from an unequal division, or whether later development made them unequal.