Most web wallets don't require KYC but are not safe while free apps like Jaxx while other apps like Xapo and Crypterium require KYC but are secure in terms of your money being safe but you cannot use them if you receive money from gambling websites (they hold your wallet's private key) unlike Jaxx.
There is no meaningful correlation between KYC and the safety of your coins. There are several big name service providers which have very strict KYC requirements which have been hacked, both for the KYC documents themselves as well as for the coins they are holding. Don't be fooled in to thinking that undergoing KYC offers you any sort of protection whatsoever - KYC only offers the exchange legal protection (or extra income when they sell your details to third parties).
Undergoing KYC to use an exchange is forgivable (although I would always recommend to use P2P or DEX trading and keep your documents safe), but you should
never be undergoing KYC just to use a wallet. It's bad enough that you are trusting an unknown third party to have complete control of your coins on your behalf. Why on Earth would you also be giving these internet strangers your personal details? Would you give your passport and credit card to a random stranger on the street for safekeeping?
The best site for estimating fees remains this one:
https://jochen-hoenicke.de/queue/#1,8h. Scroll down to the third graph "Mempool Size in MB" to get an idea of the size of all pending transactions at each fee level, and slot yourself in accordingly depending on your urgency.