This is a type of introductory clause that is called a dependent clause. It is basically one that help to provide some background information.
Adjective
adverbial clause
adverbial clause
This is a subordinate adverbial clause, as it provides information about the time ("since you left our house early"). It cannot stand alone as a complete sentence.
Adverb clause
This is a dependent adverbial clause (subordinate clause).
It is an adverbial clause of reason, as it explains why the women covered their hair.
Noun.
adjective
The clause "after the actor tripped" is a subordinate adverbial clause, specifically a subordinate time clause. It provides information on when the action in the main clause (preceding or following it) took place.
The clause "that I found in my book" is a relative clause, specifically an adjective clause. It provides additional information about the noun "book" by describing which book the speaker found.
The clause is an adverb.