Yes, FaceTime uses a data connection. As long as the iPhone has access to free wifi, FT calls will be free. If you want to go really safe, you can make sure roaming is not activated for data. That way, if the wifi drops or is weak, the FT call won’t be routed over the foreign carrier’s data network where it could incur significant charges.
I believe neither. He will be charged according to data charges. FT uses the data network, not the voice network. So it gets charged according to MBs transferred, not minutes of call or number of text messages.
Some carriers give you a certain amount of “free” roaming data. Others let you data roam for free entirely, but your speed is capped. FT uses very little bandwidth if used in audio-only mode so even an old 3G network can work just fine. Video is another matter.
If you have to pay high roaming charges per MB for all data roaming, I would advise shutting data roaming off entirely and only using these services when on wifi. If you cannot avoid costly data roaming entirely, make sure you use audio-only FT (no video) to limit consumption and also, make absolutely sure you don’t have any background processes running that could add to your bandwidth bill.
A little anecdote to drive home that point. Many years ago in Japan I had to turn on data roaming to get stupid Maps to update its cache (there was zero wifi and I was definitely lost). That worked fine, but unfortunately, Mail used to opportunity to go check for new Mail as soon as I turned roaming on and a data connection became available. It promptly downloaded a new message in the background that contained a 20 MB PDF of a technical drawing that had nothing to do with my work in Japan at all. By the time I noticed what was going on, the damage had already been done. In those days, that little stunt cost me just shy of $200. Let’s say it was a good lesson.