An important aspect of this isn't that you cant trisect any particular angle, its that you cant trisect any arbitrary angle, plenty of angles, like 90 degrees, are trivial to trisect.
Yes. The better way to phrase it would be that some angles are impossible to construct, and importantly that includes a trisection of some angles that can in fact be constructed.
Interesting. So like you have rational numbers and irrational numbers, you have trisectable angles and non-trisectable ones.
This analogy is rather precise—see this comment elsewhere in this thread.