So can I buy on an exchange and then keep it on a hot wallet online somewhere for a little while and then transfer it to a cold hard wallet? Can I transfer it straight to a hard wallet? Can I buy and sell on exchanges through a hard wallet?
You can do that. You can transfer from your exchange account to the hardware wallet and send it from your hardware wallet to the exchange site account. You must have sent your coin to your exchange account to be able to trade your coins.
if your hardware wallet already supported the dapps service and then you can do that by connecting your hardware wallet with dapps but these days ethereum dapps charge tons of money as tx fees.
Can you tell me what kind of wallet that already used by you?