yes, wallets are much safer. although sometimes you might have a wish for making some trades, try to trade scapling. small aggregators are perfect for that. try to check coinmonkey.io, it's a sort of place with many exchange rates from many exchanges.

It is a nobrainer, mate. The only founds that should be kept online are the ones that you actively trade. Then, there is no alternative.

But any founds not actively traded should be kept in your wallet and if this is a long-term state, then preferably a hardware wallet or cold storage.

The pros or cons, you ask? Security is on stake, here. If you keep your founds online you risk them, and that's it.

I am more of a long-term holder and I keep it in hardware wallet. Before I bought it, I was using wallets. It is very rare for me to keep in exchange. I don’t think it is safe enough.

From my own opinion it’s better to keep your coins in any safe wallet be it your ledger or any coldwallet available
Having  your  coins on an exchange you might risk certain things for example hacking. A lot of exchanges have been hacked of recent and this might lead to the loss of this coins you’ve been storing also your coins or tokens my stand the risk of being delisted from the exchange where you store them, and if you are not an active trader you might not know and might come back and find out your coins are no where to be found because you probably missed the deadline to transfer to another exchange.

So it’s more advisable to store your coins in safe and secured wallets instead of an exchange.

for my usual trading using the onsite market wallet but to keep it long term I would rather keep in blockchain.info wallet for Bitcoin and MyEtherWallet for ETH and other token based on ETH.

depending on the amount of storage
if you save more than 1000 USD, I recommend using wallet hardware, because it is safer
while if under 1000 USD you can use a wallet exchanger or personal wallet, both are the same ..

always be careful with everyone

I keep them in wallets and put them on exchange when needed and effective on profit. Because all put on the exchange can be evaporated quickly.

Exchanges are always subject to hacking which many believe that the owners who are behind and just made up the hacking incident. So just keep it on your wallet because the only person you can trust is yourself.

I think if you're not trading every day, it's best to open a wallet for every coin. The exchange may close, and the purse in this regard is safer.