1. "Isn't its very meaninglessness its meaning" is a confusion of reference and sense. The quoted sentence "Colorless green ideals sleep furiously" interpreted as an object in its own right holds meaning as a common example of syntax and semantics but if you "remove the quotes" and interpret its sense then we run into problems... and obviously that's what everyone is talking about when they bring it up.
2. "Is it in this way that it fails to have meaning and so will always continue to fail to have meaning?" Yep. If we assert that we're stating this sentence as an assertion or judgement of reality then we would like to evaluate its "truth". Even without getting too mystical it's a long established sense of the definition of truth that it ought to somehow correspond with reality and that any notion of truth which does not is hypothetical or meaningless. This is encapsulated in Dummett's C and discussed by Per Martin-Löf here: https://michaelt.github.io/martin-lof/Truth-and-Knowability-...
Well, the way I use words it is not meaningless. It has a pretty clear meaning. Admittedly, that meaning may not describe any possible state of the world (although I could perhaps argue that), but I can only spot that because I know what the phrase means.
Interesting. But is it meaningless? Isn't its very meaninglessness its meaning because it has come do be known due to it being bandied about far and wide as an example of meaninglessness.
Maybe it's something more. Clearly, for all X, X cannot be at one and the same time both green and colorless. Similarly ideals do not _literally_ belong to the class of things which can sleep. Finally, sleeping as an activity is never done furiously as fury signifies intent and whatever else may be true when one is asleep it is certainly true that one lacks intent when asleep.
Thus packed into that sentence are three ways in which it _cannot_ be the case or refer to something _actual_ in the world. Is it in this way that it fails to have meaning and so will always continue to fail to have meaning?
postscript:
> Let’s deconstruct this phrase and define its parts, considering the second part first.
Let's use the word analyse rather than deconstruct when that is the word needed.