The point of groups is to sync everything. If you don't want the files in the group, you'll need to delete the attachments from the group.

It is possible to have groups without syncing and personally I use some of them in that way to have the references shared but not necessarily the actual files

I'm not sure what you're referring to here.

You can disable file syncing for all groups or set them to download files as needed, but that's for controlling what gets downloaded to a given computer, not what gets uploaded. Groups are for collaboration, so again, the point is to make everything available to all devices (whether or not the files are actually downloaded on a given device).

If you don't actually want files in the group, you should delete the attachments and/or adjust the group settings to disable files altogether, in which case they won't be saved to begin with.

Again, what you want is not possible — it's not how groups work. If you don't want the files, you need to delete the attachment items, and then they'll be deleted everywhere.

Yes, because you're at your file sync quota, so no additional files are being uploaded in any groups you own, and possibly haven't been for some time. Zotero would be showing you a warning icon about that on every sync. That doesn't mean there's some feature to avoid uploading files in specific groups (beyond turning off files in a group altogether, which prevents files from being saved in the first place).