These secrets are determined entirely on a “need-to-know” basis, not the standard SHIELD clearance levels. In Fury’s parlance, it’s “compartmentalisation”.
We've seen Fury make exceptions to the privilege system in both Winter Soldier and Agents of SHIELD.
For example, when Coulson tries to access the file about his death, access is rejected despite Coulson claiming to have the appropriate clearance – presumably this was done on Fury’s orders. Or when he gets in the lift with Steve, and uses his Director Override to show him special secrets.
Fury seems to trust Coulson, when he entrusts him with super-secret projects, gives him a nice shiny plane, and resurrects him when he dies. So far, we don’t know why that is. Hill is the Deputy Director of SHIELD – unlikely she got that without earning his trust. Koenig is a wild card, but since Fury gave him a special base all of his own, he seems to be important.
Providence Base
This chosen by Fury as a place to retreat if SHIELD was compromised. If anybody with Level 6 clearance of higher knew about it, then it wouldn’t take very long to find. Indeed, as we’re told when the Hub is captured, almost all high-level agents are defectors. The only way this would stay secret is if Koenig, Coulson and Hill were specially selected to know about this base. (Which seems justified, given the Level 7 agent who sneaks in.)
Coulson lives!
The theory that Level 7s are the only people allowed to know that Coulson is still alive fails under scrutiny. There are plenty of agents with below-7 clearance who know Coulson is alive:
In The Hub, when Victoria Hand tells Coulson that “Your Level 7s can join us in the briefing”, Fitzsimmons both walk away, so their clearance is 6 at most.
According to the Marvel Wiki pages for Simmons and Fitz, this episode also tells us that they both have Level 5 clearance. (This comes up again in Turn, Turn, Turn, which I’ve checked independently.)
In Turn, Turn, Turn, we learn that Antoinne Triplett is Level 6, as of course is Eric Koenig. Of course, it would be difficult to stop either of them knowing that Coulson is alive once they’ve seen him, but nobody panics when they do.
For a large part of the series, Skye has no clearance level, and when she eventually gets inducted, it’s at Level 1.
On the flip side, we’re told in Agents of SHIELD that none of the Avengers know that Coulson is alive, but several of them seem to have the appropriate clearance:
In Winter Soldier, when Steve Rogers enters the lift with Nick Fury, we see that Steve has clearance level 8. (In the same scene, Fury uses a “Director Override” to show Steve things he isn’t usually allowed to see.)
The MCU Wiki pages for Natasha Romanov and Clint Barton has both of them listed as Level 7, although this fact isn’t cited.
Again, it seems that Fury is only telling selective people about this (and Coulson’s keeping quiet too). This is probably to avoid awkward questions about how somebody who got stabbed through the heart, and died in the Director’s arms, suddenly ends up alive and kicking.
As for the “welcome to Level 7” comment at the start of the show, I suppose that was just Coulson being dramatic for effect.
That other guy
Given the level of infiltration of SHIELD, do you think he’d want to tell somebody of any clearance level that he was still alive (especially given the number of high-level defectors). Much better to only tell the half-dozen people who he trusts, regardless of their clearance level within a now distrusted organisation.
