Is there a way to start the TCP server that will listen port using Firefox WebExtension? I know that exist way to start server with XUL/XPCOM extension, but it is obsolete and will not be supported (deprecated) from November 2017. The same question interests and about google chrome.
No, a Chrome Extension / Firefox WebExtension cannot open a socket to listen.
However, you could have a separate Native Host component that does it for you. It will need to be installed separately.