I am going to come at this from a modest level of idealism, something like Plato, rather than Berkeley.
1.) How does the idealist concede the interest of the question "Is there an extra-mental reality?" and his answer to it if, for the idealist, the very suggestion of an 'extra-mental reality' is completely unintelligible?
Our perception clearly creates the impression of there being stuff we interact with. So it is natural to assume that stuff is real. So there is no problem with motivating the question. However, what is natural is not necessarily wise.
2.) What criterion does the idealist appeal to in determining the unintelligibility of extra-mental reality?
The criterion of consistency and testability.
If the concept of extra-mental reality is at least intelligible (so as to be judged by the idealist), then what exactly is unintelligible about 'extra-mental reality' and why so?
We don't know, and we can't tell.
We know that much of what we perceive is inconsistent with itself. We see optical illusions, etc. So it is then natural to question how we correct our perception for its flaws.
Once you start trying to do so, you realize that there is no reasonable way to know when you have succeeded. You have no way of knowing what extra-mental reality is like except by comparison with previous copies of intra-mental realities. You cannot fully comprehend it, because you know your images of it are systematically flawed, and you have no standard by which to correct them.
So some part of extra-mental reality always remains unintelligible to you. Some level of your perception is illusion. And you have no idea what part that is: It could be only an insignificant portion, or the whole thing.
If 'extra-mental reality' is unintelligible not as it is considered as a concept, but as something else, how does the idealist, who concedes only the intelligibility of concepts, consider 'extra-mental reality' as 'something else' other than a concept?
There is no need to do so. If only intra-mental content is reliable, concepts are all you have. This is where Plato goes off and talks about forms.
We know that our intentions happen internally, as often they arise in a way that does not seem to affect anything else. But sometimes other things seem to respond to them. Sometimes my hand moves when I want it to. But we still can't know that this is not all an illusion.
Maybe only the intentions are real. So maybe there is not a real chair upon which I am sitting, but only the intention to be sitting and therefore something that serves the conceptual purpose of a chair. We know that the concept of the chair is real. But maybe only the concept of the chair is real.
It is clear that the world does not always meet whatever intention we throw into it. We seem to have to negotiate a lot of 'interface paraphernalia' to make any adjustments we want to the stuff that may or may not be out there. But that might all be internal resistence due to the inefficiency of our own mental workings.
Still, that seems bizarre. And we seem to evoke interdependent intentions from the environment. So either we are divided against ourselves, or there are other minds. If there are other minds, the odds are that the stuff between us is there, and that we really are using it to interact. But it is not necessarily so.
If this is not all just us, and you and I are not part of a single intellect that is segmented for some reason, then there has to be some way that intentions get transferred between minds, but it could be totally unrelated to the actual stuff we perceive, they could just be, in some sense, knobs and buttons on the user interface to whatever communication device ferries ideas between individual minds.