Thank you for taking the time to educate me on this participle beast and especially the participle clause. Up until your answer, I didn't even know they existed. I have two grammar books that don't mention them. I found this link that seems quite good-
http://www.grammaring.com/participle-clauses
It seems to me a participle clause is a short hand way to rewrite a dependent clause. They show the rewrites for most of the clauses they give. Because sometimes the subject is implied, I mistakenly think of it as a phrase.
Maybe I misunderstood you, but you seem to be calling any verb with an ing ending a participle? It seemed like in some of your examples you were calling a Gerund a participle?
Anyway, this has been very helpful because now I can understand and use participle clauses!