"rules, consequences, issues with tardiness"
These are not stage management responsibilities. Stage managers do not create sign-in sheets because we like the sound of a sharpie as it makes a perfect tick-mark next to someone's name. We have sign-in sheets because, when people flake out on rehearsals, all sorts of things go wrong: the schedule goes out the window, material needs to be covered multiple times, people miss direction, actors get pulled out of the moment (reading a line to an off-book actor is completely different from reading a line to an ASM reading from a binder and nervously twitching), company morale suffers (Either the actor gets sore about facing consequences for being late, or the other actors get sore about this actor never facing any consequences for being late. Maybe both.), the SM has to waste time on the telephone trying to find people... it's a huge mess.
This may be me projecting somewhat, but if your director expects you to take full and total responsibility for these matters (as if things like tardiness, rules, consequences, etc. are stage management busy-work), your director sounds like a wimp. I would make sure the director is invested in actually enforcing the policies and consequences he expects you to outline, because if he isn't, you may be stepping into an unhealthy environment.