Are trades done on exchanges added to blocks?
Or are they only added once you withdraw the funds to your own wallet?
If this is the case, how does the exchange decide on transaction fees?
Fees for trading are just a property of the exchange, transaction fees on Bitcoin transactions for withdrawals are just their choice to make. They might ...
You can try to import the wallet using the key you have (assuming it is a private key) through wallet software such as bluewallet.io, which supports importing of wallets via private keys.
Otherwise, if that key is not your private key, you are a little screwed, sorry.
They don't and that's the point.
The actual process varies probably from company to company, but generally, deposits into cold storage are automated, and withdrawals from cold storage require varying degrees of manual involvement.
The process usually involves some of the following components:
Air-gapped signing system
The signing device is not connected to ...