did you really write that or it an error?; I think you forgot to say I do not really know what to say. The "really" is not a problem, it's the forgetting to say " I do not"

Same question.
In English "I don't really know" means: "Actually/in reality/if I think about it further... I don't know", where 'really' modifies the phrase "I don't know" to make it milder/more casual (as opposed to being blunt), or modifies the verb 'to know' to make it clear that the speaker doesn't fully (really) know whatever's being spoken of;
or, if really is stressed: "I was only joking - I don't really know...".
In contrast, with "I really don't know", 'really' applies more to the negative don't, so it means: "I certainly/definitely do not know", perhaps suggesting that you've tried to know, but you still don't, thus you're sure and you "really don't know".

Similarly, could someone explain how the word placement in the Italian sentence affects meaning?

It's just a reinforcement, that you could translate with "just", "truly", "really", and so on; in Italian "Non so davvero cosa dire" conveys the same meaning.

Cosa as a noun is feminine, but don't be mistaken, in this sentence cosa isn't a noun (thing), but a pronoun (what) and proprio isn't an adjective (own) but an adverb (just); thus they aren't flected in any way.