How can I get the server IP address using C++?
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You generally don't want to do this. First, you may have multiple interfaces, with different IP addresses. Also, the server could be behind a NAT. Instead, the online database should discover the remote peer's address when it accepts a connection.

Do I have to open the ports I want to use in my socket? How do I open them?
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If you are behinda firewall or NAT, yes. How to do so varies with the firewall/NAT.

Can I use "default" open ports?
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There are a handful of ports that are almost certainly open for outgoing connections, like HTTP and HTTPS for example. The default on many networks is to have no open ports for incoming connections.

Hi, im also having problem with my app because of routers behind nats, i've read the article above and understand most of it, but it says at the end that the introducer/server ..."can be run without paying huge bandwidth bills for servers". This leave me with the question, if the "introducer server" cant be run by both clients behind routers, what do you use to build such a system? A Web page using PHP? I've read somewhere else that most web pages hoster wont let code run indefinitely.

Sorry if my question is not clear but it would be nice to have example of how to implement such a system.