The "arrow as improvised weapon" bit is an interesting loophole and a neat visual, but IMC it would run into the limit on the number of free actions. What limit, you ask? The one where the DM has to use his judgment about what's reasonable.
Despite the structure of the D&D turn, an archer does not fire two arrows and then stand around waiting for a while before firing again. He's drawing and loosing arrows just as fast as his hands can move. At any given instant he is likely to have an arrow in his hand, but he's trying to nock it on his bowstring; he is not gripping it in his fist like a dagger and preparing to stab someone with it.
If the archer uses the Draw A Weapon action, perhaps during a Move or with the Quick Draw feat, he can ready an arrow for melee use. But he doesn't automatically get to carry an improvised dagger around in between shots, which would remove one of the archer's big downsides at no cost to him.
Note again: This is how the DM's judgement call goes IMC. Your mileage may vary.