The password function, at the best of my imagination at the time of writing this, used for added security. URL/form spoofs are always a risk, so there's no reason why it can't be in there.
I believe removing it would mean heavy altering of the battle files (possibly all 3 of them), and template changes.
You are welcome to attempt this yourself, but be warned; I cannot provide support for errors caused of modifications to the code for obvious reasons. Always make backups
I believe you can use $owner[userid] and $opponent[userid] to generate links out of their respective names (which is which is easy to spot in the template).
I have to wonder, why would you want to generate such links, when all stats displayed in the rpg.php (apart from Battle Statistics) are displayed in the battle template itself?
If there's a valid reason for this, I would consider adding it to the next version, see