You don't have a virus. A virus is a form of malware capable of autonomously spreading by infecting legitimate files, either by appending, prepending or cavity injecting malicious code into the file. You're infected with ZeroAccess and Zbot (amongst other malware); both of which are considered Backdoor Trojans, with ZeroAccess also being considered a user-mode rootkit.
Zbot in particular targets banking information such as passwords and other login details.
If we do clean the computer as is, is there a way I can monitor the computer to see if anything comes up with the backdoor virus?
You can monitor the computer for strange behaviour. Likewise, you can monitor your accounts, etc.
But there's no actual indication. Furthermore, strange behaviour that you notice may not be attributed to malware at all.
Bottom line is - if you want to be certain your computer is clean, starting from scratch is the only way.
Is there any risk to other computer within the same network?
Impossible to say for certain, but it's very unlikely. There would be risk if any form of worm was involved, but I see no evidence of this being the case.
You may wish to reset your router to be on the safe side - I can provide instructions if you like.
If I do format, and backup files from this computer, will there be any risk with the backed up files?
I can provide instructions on how to safely backup your data.
Is there a specific procedure with reformatting to ensure there is no risk of being re-infected?
You have a recovery partition:
Partition 1: (Not Active) - (Size=39 MB) - (Type=DE)
Using this will restore your computer to how it was when the machine was first switched on.
There are alternative methods, but this is the simplest, and will achieve the same result.