Yes, Art. 112 exist.
However, you'll note it does not explicitely talk about limitation of freedom of movement.
It's an exceptional clause that is here in order for countries to be reactive to unexpected developments (e.g. immigration in Lichtenstein reaching 70% of the population (the UK's is just above 12%), financial collapse in Island).
It's not exceptional, it's safeguard. The distinction is important, safeguard clauses are invoked all the time.
It is true Liechteinstein was in a worse position than the UK on some of the important factors (ie, number of immigrants compared to number of regulars). But the UK is in a worse position than Liechteinstein on other factors, that are also important (ie, total number of immigrants). If we are talking that Liechteinstein's still has it worse, perhaps that's true, but the quota that would be imposed by the UK would be higher as well.
Is the question important to the UK? Yes, it is. The extent, the countermeasures, etc, are subject to the debate, sure.
But it's not exceptional, it's safeguard. You DON'T have to have the sky literally falling to invoke it. Sky half-falling (in your opinion, which you will then try to defend whenever, and, again, I believe there are grounds for that) to the level of clouds is enough.
Yes Article 112 is in the EEA agreement but so is the principle of free movement of people. At the very least you need to acknowledge there's a conflict between agreeing to a treaty which has at its core the free movement of people then immediately engaging an article which is extremely clearly meant to be a short term measure taken in reaction to a material change in circumstances.
There is no conflict, the UK potentially invoking article 112 would obviously be discussed in the EU, no surprises here. No rogue countries sneakily infiltrating the great community, then stabbing from the back. As I said, safeguard clauses are invoked all the time - in circumstances some parties would obviously deem completely unexceptional.
Arbitration is there to prevent abuse. Well, you are saying to me that the UK would have to make their case for invoking article 112 / defending the invokation, and I agree - I think the case can be made - but it goes the other way too. Those who would like the UK to cancel whatever measures it will impose will have to justify that. And if such a justification is successful, the result can totally be just the relaxation of limits.
Regarding article 112 not being for long-term - yes, I agree. But if the UK succeeds at making / defending their case, there is nothing stopping the EU and the UK from making the provision semi-permanent. That has been done with Liechteinstein, as I said. (And, again, that Liechteinstein is a small country has nothing to do with it, as far as EEA language is concerned.)