For me the most secure exchange is kraken. The best in terms of fees is Coinbase - for withdrawal it is almost 0 fee. The worst exchange without any doubt is Bitstamp. The fee is enourmous 0.005BTC. And each week they send me new emails asking insane things, to prove the origin of my bitcoins, fiat, by sending bank and credit card statements, pool statistics, and many other things. I stopped using this hilarious exchange several months ago.
I used bitstamp 1-2 times in 2016 and I STILL get emails from them asking to finish setting up my profile and send them additional info like what you describe.. Hilarious, indeed!
I opened an account with them in 2013, they closed it around 2016 because I lived in Hawaii at the time. Fast forward six years later and they still send emails asking for info updates even though I can't even log in.