I'll preface this with saying I don't know anything about scripting. Or at least very little. I can generally figure out what is going on by reading a script, but I can't write them at all, and ...

This may seem like an odd question but is it possible to write a script that given a set of computers the script will login locally to each computer with a specified domain username and password ...

You can customize the startup functions of your Windows operating system so the Powershell Script application runs every time you log on to Windows.