Does changing the password for Apple ID make iCloud and Find My iPhone stop working?
You need the original Apple ID to deactivate the Find my iPhone function on the phone.
If the person who has it has put it on DFU mode, and its iOS version is 7 or higher, he will need the original Apple ID to reactivate it.

You may wish to put the phone on "Lost Mode", that will put a message on the phone lock screen with information of your choice.  

It's important to know that downloading the Find my iPhone app is not the same as enabling Find my iPhone.  

There is a detailed explanation of the functionality of Find my iPhone on this article on Apple Support