Hey, Kathie- it's wonderful to see effort being put into properly rendering Native names; as someone who's worked on profiles for some of the Coast Salishan-speaking peoples, & has studied some of the languages, I've been hoping to see more Native languages represented here on WT.
I won't claim to have great familiarity with the language, but I do have a dear friend who's Oglala, & has studied the language, & I do understand a bit of the basic linguistic structure, so I'll attempt to render some of these names as best as I can, & I'll consult my friend later for more. Of course, if & when anyone with more expertise responds, I defer to them.
Scarlet Woman: Wíŋyaŋ Lúta, or Wíŋyaŋ Šá (which would be red in general- this distinction is drawn in the Lakota dictionary, but I can't be sure that the two words would have translated the same in the 1800s). Interestingly, when looking to see if I could find any evidence of the name online, I came across an account of a woman named "Tipidutawiŋ" (Dakota dialect: l becomes d), whose name would have originally meant "Red House," but was changed to "Scarlet Woman" by a white priest. No reason to believe it's the same woman, but it was certainly interesting to read about.
Light Hair: this was a childhood name for Tȟašúŋke Witkó (Crazy Horse), for who it was rendered as Žiží, which can simply be a word meaning "to be fair-haired." I don't want to make too many assumptions, but it suggests to me that the word 'hair' (perhaps Pȟehíŋ) is implied, & thus doesn't have to be part of the name. As such, Žiží.
Snow on Her: I'm not altogether certain on this one. There are a couple verbs that could be used, wáhiŋhÁŋ & ičámna. To indicate 'on,' these would be prefixed by "a-," & since the pronoun likely wouldn't be indicated, this would give us either awáhiŋhÁŋ or aičámna. The latter refers to heavier snow, but I'm not sure which would more likely have been her name, if not something else.
Red woman: based purely on the modern Lakota dictionary's distinction, Wíŋyaŋ Šá; however, it's quite possible this nominal distinction only exists in English as a means of differentiating between the two women, & that their names would have been identical in Lakhota. I am not certain.
Oyatewayankapi: Oyátewaŋyáŋkapi seems the most likely rendering in the modern orthography. This may be separated into two words, Oyáte Waŋyáŋkapi.
Tašinátópiwiŋ: I'm not 100% certain what the two morphemes at the beginning & end of the name are, but šiná is robe & tópi is four.
Will give the children a try in a little bit.