Well there is a difference in the approach, and it depends on which linguistic aspect one is adhearing to. Adverb, Verb, Preposition, Pronoun, Noun, etc are all Parts of Speech and you can read more about the different types here. http://www.arts.uottawa.ca/writcent/hypergram...
If you were to get really technical about adverbs, they can be broken down into many smaller categories called adjuncts. That is not my focus here and I am taking the collect-all aproach to adverbs, and with that approach comes the various types of adverbs of which I could summerize:
IN THE FIRST SENTENCE "She came SCREAMING into the room" the participle fulfills a progressive aspect rather than an adverbial usage. Another example may be - To go RUNNING out of a building, or, he went FLYING into the pool. This is my fault, and I now realize that that IS NOT an adverb.
BUT IN THE SECOND SENTENCE "She came into the room SCREAMING" The participle directly answers the "in what manner did she come into the room?" question and is therefore an adverb. And mind you, not all adverbs end in -ly, Such as words like "well" in "He works well with others."
As for the last two sentences I am still unsure about their role and effect on the verb, but know I would be on the right track if I were to call them Participle Clauses, or -ing Clauses.