Not if you have a "legacy" account.
I made my account a while back, and except for my email +real name + phone number, they didn't ask for any verification.
I can withdraw up to 2000 USD a day with it.
Same goes with bittrex, but i think theirs is even unlimited.
Sadly all of my legacy accounts are locked with 2FA codes that are lost long time ago.
The transaction fee is tempting but the need for ID is pretty suspicious because many exchange do not demand that at all.
I think poloniex by far has the cheapest withdrawal fee. Now, if you don't want to submit any personal documents. Your second choice will be hitbtc which charges 0.00085
BTC. Its still lower compared to others which charges .001
BTC, some are even higher. I'm not sure why is it 0.0004BTC on yours. This one is from their site:
The full list of withdrawal fees is available in your Account. Here follows the brief overview on some of our fees for popular currencies.
BTC: 0.00085 BTC
I assume you don't have your 2fa recovery code right? Maybe you can contact poloniex support on that? However their support are not that fast on responding queries but you can try your luck anyways.
Don’t have a back up code/ recovery key?
If you do not have a backup of your 16 digit key, please open a support ticket at Poloniex Support for assistance. If you're new to this, please refer to the article on how to open a support ticket and what information you should include.