Let me refine the question a bit lol. Can I use any exchange that I have BTC and ETC on and The hardware wallet supports BTC and ETC?
As long as the address is supported. Sometimes wallets can adopt some new address standards that the exchange does not yet support, but usually you should be able to find old address formats too.
when I give my crypto exchange my public address of my wallet to do a transfer , does the exchange know from the address any info like the wallet I am using....or?
No, they don't get any information from this other than what is publicly available on the blockchain. If you put in an address that has funds or had previous transactions, of course they will see them as everyone else can see them too.