I'll just start with an example sentence: The highlighted clause is an example of a clause used to modify the subject ("Runt"). In this configuration I believe it would be called an "introductory clause". But what would it be called in this, also [mostly] correct, formation: (I realize it is actually a dangling modifier, because 'her' could point to either Connie or Runt, but I'm working with a line from a classmate's story and I'm attempting to preserve the structure here). Would it be called an 'exital clause' or something like that? The author is using this formation quite often and I'd love to be able to tell them what it is called.
The second part of the sentence is participle clause, because it adds additonal information about the main subject of the sentence (Runt). In the first sentence: Though the first part of the sentence is an introdutory clause, it is also called a perfect Participle, since we know that gazing is a clause that take place before the action (took) takes place. The sentence would become a particle clause if it was written like this: Gazing down the street, Runt nervously took hold of Connie's hand (we can say that runt hold Connie's hand, and we may know that Connie is the one who is nervous). I'm not going to gurantee you that this is correct, because I myself am not good with Engish.
I'd say that it was a progressive (or imperfect) participle, because the gazing continued through the action. The perfect participle would be something like "Having gazed...".