Q: How to find out the base URI, to prepend to /api/<endpoint>?
A: You already provided us with your base url to the ticket system. /api would go after that.
Q: The IP address required is of type IPv4 or IPv6?
A: We use IP v4. In general osTicket when it asks for an IP is v4. I do not know if the application cares though.
Q: Is the IP address required a public-facing one or the server on which the application is hosted?
A: That would be up to you. If you want it to be reachable outside of your network then it would have to be a public facing one. If not, then no.