Just a few tips....
I've not been able to get into some workstation's BIOS without using PS2 keyboard/mouse. That is a more primative interface into BIOS so sometimes it works when USB KB/mouse won't. I see that you got that advice from DGroves.
We have posted over and over in here about updating BIOS from within BIOS. Search that out. I checked your workstation from my Z400 here running W7Pro64:
TITLE: ROM Firmware for HP Compaq dc7900 (786G1)
VERSION: v01.27 REV: A PASS: 1
DESCRIPTION:
This package provides the HP Business Desktop System Firmware Package for supported models running a supported operating system.
PURPOSE: Critical
SOFTPAQ FILE NAME: SP73584.exe
So, your latest BIOS is 1.27, released 11/27/15 by HP. Sounds "critical". Some BIOS updates need to get to a certain low level value before they can take the jump to the latest BIOS. Sounds like you might have a very old version but I see no place in CPUz that tells you the BIOS version. You see that during a restart briefly and can also see it once you get into BIOS.
I downloaded that latest BIOS 1.27 SP onto my workstation and ran it, cancelled out of it. There in root C/ is SWSetup/ and the SP73584 folder. Inside that is a DOS Flash folder. Inside that is a file called 7G1_0127.bin (I have my Control Panel Folder Options file tab set to show extentions of known file types so I can see the .bin part).
OK.... copy that to a fat 32 formatted thumb drive.... not too big... say 8GB. It will be the only thing. You don't try to boot into that.... you just have that in the workstation in a USB2 port and boot up, and get into BIOS, and navigate to that USB stick.
But, you need to be able to get into BIOS and navigate via BIOS to the first part of BIOS where you can update system ROM.
If you do not have these basic skills you really need to pay the money to a local computer shop to do the work for you. You are not dealing with a trivial project, and don't feel you need to know all this stuff.
But, do get professional help.....