Does anyone know if it is possible to use this editor as my WordPress editor and not the default. If so, how?
Yes, It is possible ... there is a good article on DigWP - which points out one warning that MarkDown is not reversible (ie articles written in Markdown get saved in Markdown so if you ever turn off MarkDown, then you're left with goo on your screen).
There is a MarkitUp plugin on the WordPress.org ... While the project has a warning about its age, I think you could find it still works.
Have a search on WordPress.org there are other markdown plugins
And a note from me about compatibility - you may find it easier to just use a MarkDown editor on your computer / tablet / whatever and then just export / cut n paste the finished product into your WordPress post or publish with a blog editor tool like MarsEdit.