I am a newbie and here's my first question - how do I install the Birthday by Month extension in my CiviCRM? It is not showing up in my list of available extensions. I am on CiviCRM 4.7.31
Birthday by Month extension has not been updated to work with latest CiviCRM versions and if you look at the source code here, it's only compatible with 4.2.
On a general note, extensions can be installed via CiviCRM Manage Extensions page, but that list is limited to native (CMS-independent) extensions and subjected to reviews. However, you can manually install an extension by copying the source files to the respective extensions directory (dependent on your directory settings).
Please refer CiviCRM documentations for further information https://docs.civicrm.org/sysadmin/en/latest/customize/extensions/
If you want an extension that will list birthdays in 4.7 then 'Systopia birthdays' works. It can be found here I've used it in 4.7.30 so it should be OK