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If facebook chat is turned off, can people see your activity (what you are liking etc) in the right hand side?
Yes. Even if your chat is turned off, your activity such as comments,posts, liking etc will appear on the news feed of people present in your friends list.
The option to disable the chat for some or all of your contacts should only affect your status (the "online indicator"/green dot) for them.
Required, but never shown