Is FaceTime free?

Depends entirely on the connection. FaceTime itself is a free service, so if the wifi or cellular connection your are using is free FaceTime is free. If that connection costs you or uses data against a plan quota, then those charges or limits still apply. Same thing goes for iMessage Which is also a free service, but the data connection is independent of the service. The only "cost" to either service is whether the data connection itself costs you to use.

Yes

There are no additional charges other than internet connection fee (WiFi or cellular) for both parties. If all parties are using public WiFi then it is 100% free.

....anywhere in the world?

There are some countries that have banned technology like FaceTime. This Apple article:

Wireless carrier support and features for iPhone in the United States and Canada - Apple Support

shows by regions services carriers can provide in those markets. Example: Click "Middle East. You'll find FaceTime unavailable in Saudi Arabia.

No charge for no service,