I'm sure his powers would stay- they are his now- not his by tapping through Vold- I'm sure the "others" are animagus, plus some funky spells-
going back a bit- I'm sure that a wand can't tell what a wizard will become, or no evil wizards would get wands- I'm sure they just sense power- and Voldie and Harry are both powerful- and I doubt Fawkes has any say in who gets his feathers-
also to note- the SORTING HAT is always a good key- its the one who said that Slytherin and Gryffindor were friends but Slytherin split off from them and went away- I'm still guessing something with this break is the reason for such crossovers in powers and looks with Voldie and Harry- maybe Slytherin and Gryffindor's kids married or maybe they were cousins- but the similarity in looks (also with James) is just too odd and brought up too often