I’m running sonarr on a synology for a while now, and it does have its quirks.
Permissions on synology are “weird” in my opinion. When you create a user in the synology GUI, it will be added to the group “users” automatically, and there’s no way to get them out of there (using the syno GUI, that is).
When you install a package like sonarr/drone, it creates its own user (still called nzbdrone for now). This user is not added to the group “users”, and you start running into all kinds of permission issues as you describe above, while you’re thinking “hey, I gave ‘users’ full read/write access to the folder”.
I fixed this myself by creating a group called “downloaders” in the synology GUI, granted that group read/write to all relevant folders using the syno GUI (e.g. I have a folder “downloads” with subfolders complete, incomplete, … and a folder “Media” with subfolders Movies, Series, Music, …), and then, the most important one:
Log in to your synology over ssh (e.g. use PuTTY or something on windows, if you’re using a linux flavor you’re laughing in my face over the fact that it’s almost 2015 and Windows still can’t do ssh out of the box) and edit your /etc/group file. On the relevant line, add all users for the download tools you have. For example I’m running couchpotato, nzbdrone/sonarr, sab and transmission:
downloaders:x:65543:couchpotatoserver-custom,nzbdrone,sabnzbd,transmission
This is not for the faint of heart, and I really really recommend that you make a backup of the group file, and I’m not responsible for damage etc etc.
I think there’s a couple of other synology guys/gals here, maybe they’re doing something different.
I know at least one person has just created a separate user via the synology GUI, and then changed all relevant scripts for download tools to “run as” that user. e.g. start/stop script for couchpotato, drone, sab, …, update scripts, install scripts, whatever. You’ll also need to change /etc/passwd for this approach if I’m not mistaken.