Let's say I want to make a large transfer to an exchange. I am sure to verify the receiver address. However, if the exchange made a mistake somehow, all the bitcoin would be lost forever. So does it make sense to send a small amount first, confirm that they receive it, and then send the rest?
lol, this is not really necessary, unless you are a very careful person 
BTC is not like other altcoins, BTC works on a single network and just make sure you send it to the right address, it never goes away. It's not a good idea to do 2 transactions for safety, because it costs too much per transaction, I don't want to spend more Because I have also heard that each address should not be used more than once.
Where did you hear this from? Personally, I think it's safer to use the same address more than once, since you already know that you can successfully deposit it.