If you would like to have Firefox on your mobile device, you don't have to have a desktop computer at all. It looks like you have it successfully set up and working.

Now, if you wanted to use Firefox Sync, yes you would have to have a desktop computer. Basically Sync allows you to synchronize bookmarks, passwords, history, etc between different computers and mobile devices that have Firefox on them. If you only have one device, there isn't any real need for Sync.

You can read What is Firefox Sync for more information if you'd like.

Or maybe I'm horribly misunderstanding your question, so please let me know.

There is no requirement to use Sync to use Firefox mobile. Just ignore it as it does not apply to you.