Good morning, Yuriya.
(1) This is such a difficult matter that some books no longer use words such as "gerund" or "participle" anymore. They just say -ing words.
(2) Of course, I dare not give you any answers.
(3) All I can do is to offer what I have found in my favorite books. Then you will have to decide for yourself.
(4) In Descriptive Grammar (Professors House & Harman), there are some sentences that seem similar to yours.
That fellow kept gritting his teeth.
He felt himself sinking.
We saw them eating peanuts.
The professors say that "gritting" is a participle that modifies the subject "fellow"; "sinking" and "eating" are participles that are being used as objective complements of "himself" and "them."
(5) In A Grammar of Present-Day English (Professors Pence and Emery), they analyze a sentence similar to your last sentence:
The children came running into the house.
The professors say there are two ways to analyze this sentence:
***** Method #1.
"Running" is a participle that refers to "children"; "came" is being used as a linking verb.
***** Method #2.
(a) This is very difficult.
(b) Many (many!!!) years ago, the English people used the preposition "on" + gerund.
(i) So maybe they would say: The children came on running.
(ii) Then the people started to say it faster, so they started to say: The children came a-running.
(iii) Finally, they dropped the "a," and it became Method #1.
(c) Some experts say that -- if you wish to -- you may imagine that there is a preposition there ("on" or "a"). In that case, it becomes a prepositional phrase that modifies the verb "came."
Therefore, we can say that "running" modifies the verb came.
(d) Today people still occasionally use a- + gerund to give their speech or writing a different flavor:
She burst out a-crying./ Hurry up! Time is a-wasting!/ I was a-wondering what had happened. Thanks for finally telling me.
Finally, to show you how difficult this is, here are two sentences from Understanding English by Professor Paul Roberts:
(a) I saw HIS walking. (gerund -- noun)
(b) I saw HIM walking. (participle)
Professor Roberts says that (a) refers to the way that he walks.
(My example, NOT the professor's: Look at his walking. I think he's sick.)
The professor says that (b) refers to the fact that he is walking. (Again only my example: He says that he cannot walk. He is lying. I saw him walking yesterday.)
***** Have a nice day!
Student or Learner