I have a Bitcoin wallet through an exchange, can I have a wallet not attached to an exchange? or multiple wallets? if the exchange has problems = I have problems 
Yes, and you're right its the same if you have a web wallet so I recommend that use a mobile wallet like Mycelium, desktop wallet like electrum or a hardware wallet like ledger nano s tho and trezor tho you have to buy the hardware wallets first. I'm assure you that this wallets are the safest one. Look here https://bitcoin.org/en/choose-your-wallet