Hey Munda,
He ran into the building,determined to save his puppy.
"Determined" is modifying "he", so this would not be a preferred way to write this. We'd prefer to see the participle touching the noun.
You seem confused about the difference between a participle and words with -ing. Words ending in -ing can be participles, gerunds, or participles that are part of progressive tenses.
PARTICIPLES
Present participles are generally adverbial when they follow the comma after a clause.
I crossed the pool,swimming with all my might.
In your example, "swimming" is iffy. You could argue it's modifying "crossed" (the way in which the crossing occurred), but it's really modifying "I". Just like the last one, it's not the cleanest way to write it.
The car going down the street is red
In this version, there's no comma. Participles with no commas always modify whatever they're touching. Very straightforward.
GERUNDS
Participles and gerunds are totally different words (adjectives versus nouns). They may look the same, but they're homonyms, with different meanings. Gerunds are nouns, not adjectives.
PROGRESSIVE TENSES
I was running. (this is really a participle with the past tense verb)
I am running. (this is really a participle with the present tense verb.
The leader came in a car,with his bodyguards following suit.
"following suit" is an idiom meaning "doing the same". This is just a participial phrase modifying a noun (no comma), "bodyguards."
Hope that helps!
-t