Is a dependent clause part of the superordinate clause's predicate?

Yes, according to the traditional definition of "predicate", the when clause is part of the predicate of the main clause, just as any other adverbial phrase of time, like during the spectacle.

But each clause can have its own predicate, as it is the case here; the subordinate clause has a subject (I) and a predicate (see closed doors) too. One clause can be nested within another clause while still having a predicate and a subject of its own; consequently, the words see closed doors are both part of the predicate of the main clause and part of the predicate of the subordinate clause.

The conjunction when, which governs the content of the subordinate clause, is normally not considered part of the subordinate clause itself, in that it is neither part of its predicate nor of its subject. (Note that, in relative clauses, the relative pronoun is part of the subordinate clause, so that it can be the subject.)

Following Jespersen (The Philosophy of Grammar, 1924) the Cambridge Grammar of the English Language takes when to be a preposition taking clausal complements. Thus, when I see closed doors is a preposition phrase. The function of this preposition phrase is that of adjunct in clause structure.

CGEL also uses the term PREDICATE to denote the head of a clause, a function filled by a verb phrase. The adjunct is neither part of the subject nor part of the VP functioning as the predicate. The syntax tree, then, looks like this: