Questions Regarding Binfinex, Poloniex and Transaction Fees

Hi All,

I started studying cryptocurrency from last week. I've been tempted to make a Bitfinex and Poloniex account however from what I notice there seems to be legal issues with Bitfinex recently and frozen transactions plaguing Poloniex. Wanted to ask a second opinion if these exchanges are safe and reliable and if there are any grounds to the legal claims made against Bitfinex.

Also regarding transaction fees, if I were to transfer BTC and Eth from one exchange to another or from wallet to exchange. I understand there are deposit and withdrawal fees where I need to pay withdrawal fees to the exchange i'm withdrawing from and paying the wallet/exchange which I'm depositing in if I understand this right?

Lastly apart from those fees, I heard each crypto also has fees tied to the blockchain which I have to pay on top of that. When does this happen? When we do trading in the same exchange or when we move money from one exchange to another?