On my analysis it is not a clause, only part of one. Some people do treat it as a clause, and then it is a main clause -- it's not dependent on any other element in the sentence. I think it’s better to say that in your example the sentence as a whole is the main clause.
I very much dislike the term 'noun clause'. The classification of subordinate clauses should be based on their internal form rather than spurious analogies with the parts of speech. The that clause is thus a declarative content clause functioning as complement of "imagining".
This is a relative clause. See 2) above for why the term 'adjective clause' is best avoided. Note that not everything that modifies a noun is an adjective.
Yes, but a relative clause: see 2) and 3) above.