Unless your account was created before they changed their policies (long ago) it's impossible to trade without passing the verification process.
I have a legacy account there (unverified) which I created several months ago and it's limit is $2000.

There's no deposit fee's from what I know but some coins has a minimum deposit. Withdrawal fees in bitcoin would be 10K sats.
As for trading fees, You can refer to this: https://poloniex.com/fees