There are some different way you can achieve many of these. There are like the cloud collaboration suits, and syncthing way

I want to sync my calendars and contacts

For this you can have something like nextcloud or it's alternatives, or syncthing with decsync, or a separate caldav service

I want to download media like TV shows and movies. (And music, too). “Of course, only legal obtained from the internet cough.”

I personally use jellyfin + transmission. I'm still trying to set up *arr suite, but it's not working, then I could use something like jellyseer. But transmission is working well anyway

Is it custom build NAS or something like Synology?

As already mentioned for proper backups you should locate the NAS somewhere else.

I'm not an expert by any means so don't take my word for it. I've tinkerd with some pies befor but this was my proper first project: a NAS for my family (multiple households).

If it's for personal use, you can use a VPN to your local network. It's by far the safest option and easier to setup.

I went with OMV (openmediavault) for local sharing as OS + Docker + portainer. For the backups and access I went with Docker Nextcloud and Docker Swag (letsencrypt) and a reserve proxy with duckdns since I don't have a fixed IP. Nextcloud setup was the easiest part and not really a hassle. With a VPN you won't need to tinker with the "access from net" stuff.

For syncing files on Android I would recommend Autosync since it's the most reliable one IMO. No other OS tested so far.

Nextcloud can handle a lot on its own and for something special you will find a proper plugin. For the download needs you may find some docker containers.