HP's line of ThinPro thin clients fulfills almost all of these requirements. We considered others, especial the Wyse ThinOS units; but they topped out at two monitors. We really wanted to avoid Windows based systems for a variety of reasons, not the lest of which were cost and vulnerability concerns. The ThinPro units were notably lacking SNMP support, but fulfilled all our other requirements. Additionally I have previous experience with these units, so I am a bit more endeverous than I should be (they are incredibly difficult to brick, and the software HP releases might officially support only the latest hardware but runs great on older hardware as well).
HP's ThinPro operating system is a lightwight version of Debian Linux with a variety of HP proprietary software mixed in. The important fact here is that it's Debian with some extra bits. I started by downloading the latest version of ThinPro; version 5.2 as of writing (download image, or the Windows utility to make a bootable USB stick). Load up the image as you typically would. I'm using an HP t5565, an x86 based device. The same process applies to the ARM based thin clients, but you'll have to modify everything here a bit, for example downloading the T6A image.
Once you have the basic system running, switch to Administrator Mode (typically right clicking the desktop), open a Terminal (under the search menu); or SSH into the device (if you have SSHd enabled). Enter the command
Now you have SNMPd installed. But it has the stock configuration. Unfortunatly I do not have time to modify Net-SNMP to read its configuration from the Manticore Registry (the configuration database HP's thin clients use to provide a common configuration interface across ThinPro and Windows Embedded Standard devices). It appears that HP may be working on this, but do not have any inside information. For now you will have to edit the standard configuration file to get the results you want, it's path is