Guide: Running Jellyfin in Synology's DSM 7 Using Docker Compose

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I have my music folder on Synology, and then i have sub-folders regarding the different music styles and the another subfolder (inside of the genres one) for each artist.
Now i can't remember if i had to give the path to each individual folders (ex: - /volume1/musicNu-Jazz Cool, Jazz, Chill-Out, Funk, Soul) or just to "- /volume1/music" and then on Jellyfin Dashboard - Libraries, i have to give the path to each folder.

On Container Manager's side, all you have to do is mount the music folder to let Jellyfin see all the subfolders.

On Jellyfin's side, make sure you add each subfolder directly to the music library. So directly mount all your genres to the library.

1- When an update is released, i only need to stop Jellyfin, and after the update, restart it?

You update the yaml file to the newest version number, then build. Container manager will handle the rest. It will pull the image, then shut down the current container, rebuild it with the new image version, then deploy it again.

2- How do you access your library when you're not at home? I know that we can use VPN but i tried using with my Synology ddns (like i do with Synology apps) without success.

Check out Tailscale. It offers a free mesh VPN service (free up to 3 users) that lets you remotely connect together any device that you can install it on. Synology has a Tailscale package. There's also a Tailscale app for Android and iPhone too.

I would actually say that you should be using Tailscale over Synology's Quick Connect for remote access.