100% untrue on all counts.
Punishment is based on the frequency and severity of your behavior when you're reported. Report volume is irrelevant, as false reports get tossed out.
Manual review is unnecessary for almost all cases, as there is a path of escalation based on the frequency and severity of the behavior (as I mentioned). This means that it takes a lot more than a few games of minor toxicity (which basically everyone exhibits sometimes) to get punished; it's the pattern of that behavior that results in punishment, making the system pretty accurate for not punishing unfairly or too severely.
Riot actually does manually review all the time. You can just look at Riot Tantram's boards profile and see that he does it on his own free time, and you can see people posting their ticket discussions on the Player Behavior board as well where Player Support Specialsts like Jenny711 review the case upon request. The thing is, the IFS is so accurate with punishing negative chat that a manual review comes up with the same conclusion nearly 100% of the time; when there is a mistake, they really do remove the punishments, and it doesn't matter whether you're a high profile player or not.