Will exchanges use the LN for trading transactions?
What do you mean with 'trading transactions' ?
The moment you are trading coins on an exchange no transaction happens.
The coins are owned by the exchange at that moment. The next TX happens when people withdraw.
An exchanges uses transaction to 1) pay withdrawals 2) move funds from/to cold wallet 3) batching/consolidating.
Wouldn't they need to run very "large" channels, and be the target of hacks?
Just like they need to have 'large' amounts in a hot wallet to let people withdraw. They are the prior target of hacks regarding crypto currencies, regardeless of LN.
I know the channel needs to be funded, so where does the coins stay?
You might want to read the LN whitepaper:
https://lightning.network/lightning-network-paper.pdfWill they be able to hold the majority of their funds on cold storage, as they should?
Yes. Just like they have their hot and cold wallet now, they will have their cold wallet and fund their channel (LN hot wallet) with it.