Mine early modern English senses tingle...
V3 P1
~67%: "Hath thee rethought" -> "Hast thou".
"Hast" is the second person conjugation; "hath" is third person (he/she).
"Thee" is the second person object pronoun, equivalent to "me". "Have me rethought" would be obviously wrong (should be "have I rethought"), so the same is true here.
~68%: "Doth thee?" -- three possible corrections:
"Dost thou?" -- same as above; right now it reads as the equivalent of "does me?"
"Hast thou?" -- strictly speaking, "do you" as a sort of catch-all is a much more modern construction; for something in this register, they'd echo the verb of what the prior speaker had said (in this case, "have", since it's "I've
got something else to discuss")
"Verily?" -- if you just want to avoid that headache entirely and just want the vibe of "is that so?".
~70%: "And what doth be this request?" -> "And what is this request" -- it really is that simple; "doth be" isn't a thing.
~75%: "Though that doth only apply" -> "Though that applyeth only" -- "Doth only apply" is the early modern equivalent of "does only apply", and considering the context, I don't think they'd use "that does only apply", which is a phrasing we'd only use if we were trying to emphasise "only" as a contrast to something else. Not strictly a correction -- it's grammatically correct as-is -- but I think the nuance is off
~76%: "Your opposite, I do suppose" -> "Thine opposite" -- "Your" is for formal or second person plural, which doesn't match the otherwise casual tone thus far.
~76%: "Why doth thee look so glum?" -> "Why dost thou look so glum?" -- Same reasonings as earlier
~78%: "her personality has deprived her of any friends" -> "her personality hath deprived her of any friends" -- "Has" is 1700s+, IIRC, and not generally associated with the register being used here.
~80%: "All sorts of magic, ye say?" -> "All sorts of magic, thou sayest" -- "ye" is formal (c.f. "usted" in Spanish), and the second person conjugation of say is "sayest" ("say" would be correct for first-person, "sayeth" for third person)
~89% "... or fail to praise thy achievements" -> "or fail to praise thine achievements" -- "Thy" vs "thine" is the same as "a" vs "an"; in this case "achievements" starts with a vowel sound, so "thine".
~93%: "Oh, there doth be no need to worry" -> "Oh, there is no need to worry" -- again, it really is that simple.
Hope this doesn't come across as harsh, but this is my pet peeve :P
NOTE: if the intent with this character is that she's trying to sound old-fashioned and just jumbling everything up, ignore all of this, but that's not the feeling I get from her.