Zotero, as you've found, comes pretty close except it doesn't do markdown. You can host your own WebDAV server to host the PDF files for sharing. Just change the file path in the preferences.
I, Librarian is another that comes close. Works pretty much like Zotero except it has no dedicated desktop client. Everything is stored on the server. And still no markdown.
Maybe check out https://github.com/bpatrik/pigallery2