This is the closest I have seen to making something work, but there is no more taxi squad, and Hague won't be on IR forever

You can probably stick it out with 21 players and moving some players like Miromanov down when Hague comes back.

The cap should work by then. It's better than the cap penalties if you go over the cap!

I have no idea what the cap penalties are, I thought that you quite literally just could not go over the cap, are any of the players on the roster waiver exempt though, I wouldn't think they are as they have been playing a lot.

Closest to value yes but in reality there's too many moving pieces.

Well... they have to move 10M$ in the next 2 weeks. Its gonna require some ingenuity

Some of them surely are , Miromanov is Waivers Exempt

The simple answer is that you cannot go over the cap. That doesn't mean it isn't possible, only that there are measures to prevent most scenarios.
If something does slip through those measures, it can result in hefty fines, draft pick forfeitures, and game forfeitures.
Needless to say, teams are sufficiently incentivized to resolve those issues themselves. A team will move a player at a loss before they will allow their roster to be over limit.
Occasionally injury issues will prevent a team from having the flexibility to ice a full lineup due to cap restraints, but putting more cap in the lineup than you are allowed could only really occur as the result of an oversight. Teams have to submit their lineup for approval prior to a game.

The answer that matters here, is that Eichel can not be activated from the LTIR until there is sufficient cap room.
That doesn't mean they can wait and keep him on IR however. Once a player is cleared, they have to remove them from IR.
Attempting to keep him on IR longer would almost certainly result in cap circumvention. The consequences of which would likely be severe, given that Las Vegas was entirely aware of the situation and had/has plenty of time to achieve compliance.
There is also grounds for Eichel to file a grievance if he chooses, (though I doubt that is likely given that he has a guaranteed $10M contract that he probably wouldn't get elsewhere at this time).

Short answer: Eichel can't play until there is cap room, but if there isn't cap room by the time he is ready, there are consequences.

Thank you, that is very good to know. So essentially, they probably need to make a move within about a week to clear cap space since he has been fully cleared for contact?

We don't have the exact timeline, but they have 3 games / 6 days where they can put Eichel on a conditioning stint in the AHL while keeping him on LTIR. That will likely be the precursor to him being "activated". Once you see that happen, they are on the clock.

Main piece here for 2 cap dumps is a super late 2nd and a 25 year old 4th liner. Its a 1st for Dadanov and a 2nd for Smith to dump their cap on the Habs