You can turn on the invisible feature for all your Facebook chat options by using the following steps.
Or you can click on the Messenger icon and instead select See all in Messenger to jump to the main Messenger interface.
When you are online, you can check out how many of your friends are online and they also able to see your online status on Facebook as well.
You can block a seleted friend or an entire friend list, here is how: 1.
Use mock email accounts.