***** NOT A TEACHER *****
I'm not a linguist, either, but I DO love to talk about grammar jargon!
I shall let more qualified people than I (am) answer your questions. My contribution to your thread consists of two examples of participles being used as subjective complements (at least, according to that one book):
1. "The girl came running."
2. "That fellow kept clenching his fists."
-- House and Harman, Descriptive English Grammar (1931 and 1950).
Personally, I like that analysis, and that is how I diagram such sentences. But there is another analysis that some books prefer.
Many, many years ago, the sentence might have been expressed as "The girl came a-running." "A" is a preposition meaning "on." The word "running" then was a gerund. People then started to drop the "a" and just said "The girl came running." So some grammarians prefer to still label "running" a gerund that modifies the verb "came."
Who is "right"? Of course, I do not know.
The book that gave that explanation says this: "ften there is more than one good explanation for a construction in English grammar."
That book is Pence & Emery's A Grammar of Present-Day English (1963).
Student or Learner