F-droid app allows adding other repos. An other repo is just a HTTP server with a index.xml and bunch of apks. Since you already have a server with a bunch of apks, please consider also generating and hosting the xml file.
To upload designs, you'll need to enable LFS and have an admin enable hashed storage. More information
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IMHO F-Droid classifies as a distro, and it's up to the distribution to, well... distribute packages. I don't see why we should do that, and why this ticket needs to stay open.
However once again, I'm totally available to help them building VLC for android.