50 workstations, 1 .bat file on a network server, will it work?

This should work so long as the account you run the scheduled task using has correct access to the network share where the bat file is located.

Depending on what the script does, I'd stagger the run times by department. You only need a minute or two of variance to minimize the odds of a staggering hit right at 12. Regarding the commenter above who mentions a login script that high numbers of machines hit, computer logins do not occur at exactly the same moment. Even a few second difference between logins will result in different times for the script getting accessed, so his case, while close, is not exactly the same as yours regarding high levels of simultaneous file access.

Having batch file will work but consider limitations such as having appropriate permission and admin requirements.

You may consider deploying it as Logon Script thru Active Directory.

Why would you imagine it might error out? Each workstation simply reads the file and executes its commands. Provided your server is real server and not just a workstation pretending to be a server there will be no issue.

The only thing you need to keep in mind is that a batch file is read and executed one command at a time. Therefore, if you were to make a change to that file while it was being executed the behaviour can be somewhat unpredictable. The actual results would depend on what has changed and exactly where in the file the client is at that time.