Didn't כרגע get used to mean now in one of the other sentences? (The או אכר כך one I think)
(format) כרגע means "right now, at this moment". So sort of yes, but not exactly, you could say בקרוב אבל לא כרגע and it means about the same thing, but not quite. It's more specific.
Yes, but the English didn't specify. They should both be accepted
בקרוב אך לא עכשיו what is wrong with this
What's wrong with תיכף (or תכף)? Is it too soon (= very soon)?
Wouldn't אך be correct, too?