Is back up to a network folder possible? Is there a Backup App that would allow me to backup apps and application data to a network folder?  I don't mean to backup to the cloud (e.g. Dropbox) but to a shared folder on my PC or a NAS.

Over Wi-Fi would be cool, but I don't mind if that is via USB. I'm open to both.
If you combine two apps, this is possible. Of course, root is required on your device to backup app data, as you certainly know.

Now the first app you will need is Titanium Backup. This app not only creates a backup of all apps, app data, and even system data on your SD-Card, contains several useful tools for maintenance, and can schedule its actions -- it also allows pushing the backups to Dropbox.

But yes, I read you didn't target the Cloud, so here comes number two: FolderSync can sync your files/directories (and thus also your backup folder on the SD card) with several cloud services as well as with your own servers, using e.g. FTP, FTPS, SFTP, WebDAV or Samba (Windows Share) for the latter.

While there's no real alternative to Titanium Backup, there are some to FolderSync: Titanium Media Sync from the same author as Titanium Backup supports Dropbox plus FTP, SFTP, and FTPS. And FTPSyncX supports FTP, FTPS (SSL), SFTP (SSH), Samba, and Dropbox. I use the Pro version of the latter, and so I can tell you first-hand it works nice via Wifi -- you can even tell it to only sync when connected to your home Wifi, and run in certain intervals.

I almost forgot there's a tool that could serve both purposes, backup and sync in one: rsync backup for Android. But it may be a little more complicated to setup, and it requires ssh on the remote side -- not a realy show-stopper if your PC runs Linux :)