Firefox opens up about a dozen connections on launch

According to this knowledge base document Firefox may establish connections on startup for the following reasons:

• A loopback connection (to IP address 127.0.0.1) on non-Unix machines 1. The browser is communicating normally with itself, and it is not recommended this be blocked.
• Loading Live Bookmarks. Deleting all Live Bookmarks will stop these connections.
• Checking for application updates. "Tools -> Options -> Advanced -> Update" provides options to disable update checks.
• Loading home page. Setting the homepage to a blank page or a local page will stop this connection.
• Updating phishing sites database. Setting browser.safebrowsing.enabled to false in about:config will disable this feature.
• Prefetching links if the page it's on defines any. See the Prefetch FAQ and network.prefetch-next for further information. To disable pings in prefetched links, see browser.send_pings.
• Updating the Add-ons Blocklist. Setting extensions.blocklist.enabled to false in about:config will disable this feature.
• Updating metadata for installed add-ons. Setting extensions.getAddons.cache.enabled to false in about:config will disable this feature.
• Firefox Health Report data.

The page was last modified in 2013, so there may be more reasons nowadays.