Is this the normal practice? What if you created a wallet in one of the local exchanges, and that local exchangers doesn't have the option for you to change your bitcoin address? Then what should you do?
Set aside a dedicated wallet just for using with the site in question. You dont need to be paranoids its fine to use the same address.
Why would people want to know your activity anyway ? Even if they know so what ? They can do much with that info. Just make sure your personal details are not bundled together with your wallet or people will know that it is you who is using that wallet and Eben then who cares ?