Some people are designated by Yahoo to be "trusted members of the community." I happen to be one, but don't ask me how that happened. Anyways, when I report something, it gets deleted immediately. I try to use that power judiciously and sparingly.
It does not come automatically with getting to Level 7. Some claim to be "trusted users" who are Level 5 and Level 6. No one really knows how that "trusted member" status is bestowed, but some will authoritatively repeat a lot of theories on relative trust levels and reporting frequencies and this and that. Personally, I acquired that status after about 3 years on the site, at around 50,000 points, and having made less than a dozen reports...so who really knows?