Now in your eyes, is this a flexible, secure design? Or is it naive? Am I missing something? Is it too much?
It all depends on how critical your system needs to be. For all uses except the most secure ones (government, large corporation, easy online targets, etc), this is a bit overkill. I would personally recycle the firewall/router and put either the application server or directory server apps on it. It's secure enough, and otherwise that machine's gonna be pretty idle.
Also, I second BashLogic. You do not want the file server (or even the printers) through wireless.
If the printers don't have Ethernet ports, plug them with USB on the firewall/router and share them, or your could get a small printer server.
You might also consider combining application and directory server if you can. I don't think there would be a security issue, and nowadays machines are pretty powerful and can easily handle multiple services.
I could argue the need for the backup server, but it might be a requirement. If you have a diligent IT, this could be overkill. You could have a large RAID backup along the application server instead, with scheduled backups, for an inexpensive setup. What's way more important is the way you configure your RAID setup, and the way you backup. I sometimes make have two separate backups, on isolated media.
I would personally avoid any portable backup devices (blu-ray or tapes). I favor redundant removable HDDs. Capacity-wise, maintenance cost-wise, it's very relevant, risk-proof and future-proof.
My 2 cents.
