As far as I know there is no command to list all ports that mIRC is using, but if there is I'm sure someone will post it. As you said, the connecting ports are usually 6667-6669 and 7000, however, different servers can have different ports to connect to. Type
//echo $port to know which port you're connected with. To know what ports are needed to connect to a server try typing
/motd and looking at the servers Message Of The Day. If it does not say, it should tell you the admin of the server - try emailing them to find out. 97% of servers will work with 6667 however.
By default DCC ports are set as 1024-5000 in your DCC Options:
ALT+O > DCC > OptionsThese can be changed of course to work with your router. Any ports, preferably consecutive, in the range of 1024-5000 will be fine (for example, first: 4440 last: 4441).
IdentD uses port 113 - if port 113 is closed on your computer you will see a ~ prefixed to your Ident (/whois yourself to see it) OR you have not turned the IdentD feature on. Use
/identd on to do so.
Happy chatting

Regards,