There's an argument to be made that there could be a BTC supply shortage on exchanges prior to August 1st, due to a larger percentage of people pulling their bitcoins off into wallets they control. If that happens, but demand remains constant, you could see the price go higher. I'm currently bearish, but there might be an opportunity there for people willing to take the risk of keeping their bitcoins for sale on the exchanges nearer to August 1st.
Keeping your coins on an exchange may be your only chance to capitalise on multiple chains. It's quite possible one or more will be so painfully slow until a difficulty change that you won't be able to move them quickly enough. By that point it may all be over.
That's true, too, as long as the exchange provides you with virtual bitcoins on both sides of the fork. It's not clear to me that every exchange will do so
Coinbase will certainly refrain from that
As I got it, they basically stole Ethereum Classic from their users after the Ethereum split, so there shouldn't be any doubts that they will do it again with Bitcoin (though they may have hard time this time given they are going to support the hardfork). Major exchanges (like Bitfinex) will likely create wallets for the new coin when the dust settles a little. Regarding moving coins from exchanges to personal wallets, I'm more inclined to think that most traders will just sell their stashes for fiat or other coins such as Litecoin or Ethereum (I opted for Litecoin myself), and that will likely crash the prices on its own (this might be what we already see commencing)