He may have been punished. If more than one of you reported him from the match, only the one who submitted the last report would see the pop-up alert about him being punished. But just because you don't see it, doesn't mean a player hasn't been punished. Also, if he hadn't been issued a previous punishment, then he may only be given a chat restriction. So, even if he is punished, and you are stalking his match history hoping for a ban, and see he is still playing, it doesn't mean he wasn't given any punishment.
Currently, the IFS responds to reports within approximately 15 minutes though notifications may be delayed due to players hopping into another match, logging out for the night, etc. And as far as the punishments for false reports Riot takes that very seriously. "Sometimes a player attempts to punish another player by reporting them as a form of retaliation. In a system that relies on honesty, that’s potentially a big problem. We can detect false reports, filter them out, and decrease the effectiveness of future reports from consistently dishonest players. As we continue to build more systems that rely on honest reports, the penalties for false reporting will become more strict." Currently, groups and premades are not allowed to gang-report other players. The system is coded in a way that doesn't let their reports have the same weight as the same number of strangers would. And as far as players who consistently report players who do not get reported by other players.... The system tends to toss those reports, unless there is overwhelming evidence of their veracity. For many of the players who file so many reports, they simple cease to be take seriously.