When a creature is summoned but still on the stack is it considered a permanent?
It's not a permanent while it's on the stack.

110.1. A permanent is a card or token on the battlefield.
111.1. A spell is a card on the stack.

Before it resolves, it's a spell (on the stack). After it resolves, it's a permanent (on the battlefield).
So no, you cannot cast Avoid Fate targeting a Counterspell that's targeting your Elvish Mystic on the stack. (It's not that Avoid Fate has no effect; you can't cast it in the first place.)