[deleted] with the Direct Links extension you could point your users to /composer
, which will automatically show the discussion composer.
You can also use that extension as an example how to open the post composer. Opening automatically on the homepage is a bit trickier though. Here's how FoF Terms opens a modal on the homepage under some conditions: https://github.com/FriendsOfFlarum/terms/blob/master/js/src/forum/addAcceptModal.js