Every new founded city-state comes with three special facilities: Energiser, Research Lab and Constructor. These are the default facilities which can't be demolished. You can build only one additional special facility, besides the default ones.
Before you can build any additional special facilities you have to research the corresponding improvement. Only then that facility will appear in the construction list.
Researching the same improvements increases their power which takes effect immediately. For example: researching tax collecting allows you to build tax collector. Researching the next level of tax collecting will automatically improve the existing facility's effectiveness.
If you demolish a special facility, you lose the provided effect. May you build it again, the facility will provide its bonus at the current researched level. Thus, if you demolish the tax collector while continuing to reasearch the tax collecting improvement up to level 5, when you build the tax collector again, it will provide you with the level 5 tax collecting.