In the chat-options, you can select "offline".
You can 'hide' someone from showing up on your live feed- you click the lil x in the upper right corner of their post and have the option to hide the person, mark as spam, or cancel. However, even if you hide someone from your live feed, you can still see whatever they post by going to the person's profile directly. As far as I know, theres no way to find out if someone hid you on their feed unless you ask them.
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