External HDD shouldn’t be a big issue one way or the other for any of these solutions. The more relevant question is how structured the data is (and whether that matters) - are your torrents downloading to separate movies and TV directories, which can just be mirrored across, are your series downloading as individual episodes, and if so, do they need to be restructured to series/shoe/season/episode or similar?

VNC/RDP are remote desktop solutions

I can’t help with the script, sadly… But I’d do a general search for “Linux script to move files on a scheduled basis” (or similar) - there will be guidance around rsync/rclone floating around. Filebot may be another option.

I believe the *arrs can be configured to pick up from a folder on a scheduled basis - they can certainly do it on a manually triggered basis.

You can probably connect to the server outside your home, though the specifics will vary depending on what you want to do - you’ll likely need to forward ports and use a solution like duckdns, or (preferably from what I hear) configure a reverse proxy like ngnix. Port forwarding may be an issue if you’re using your ISP’s router.

For my part, I use usenet rather than torrents and the *arrs, which organise the files and push them to plex, which is shared remotely - I also have external access to the *arrs via port forwarding and duckdns, and a VM I can connect to via my server management page (unraid).

Probably start by taking a look at Filebot.