Why is Fluoride anion less stable than iodide anion?
Actually, the answer is a bit more nuanced. If we observe the pKb values (i.e. tendency to form HX from X-, where X is a B-L base) it is pretty clear that F- would form HF very quickly. 

The explanation is that I- would have a much larger area to balance out the negative charge compared to F-, and the anion would be more stable as a result.