So does it make sense to send a small amount first, confirm that they receive it, and then send the rest?
If you're extra paranoid, sure. But exchanges not receiving your deposit are really rare cases(assuming the exchange is actually legitimate), and mostly occurs through obscure altcoins. You're really not likely to have a problem when it comes to Bitcoin.
Because I have also heard that each address should not be used more than once.
This is mostly due to privacy reasons with your own wallets. As for exchanges, I'm not sure what exchange you're using, but if I remember correctly all the exchanges I personally use don't change deposit addresses. They only change when I request a change.