The question seems to be about the difference between gerund phrases and restricted relative clauses.
The work breaks down into sprints lasting three weeks each.
The sentence can also be written as:
The work breaks down into sprints that last three weeks each.
The first sentence uses the form of a gerund phrase that goes with sprints.
The second uses a relative clause. The semantic meaning is the same. Therefore, ít is useful to try and write you sentence with a that clause and then see if it makes sense as a gerund. In this example, the gerund form is for an action verb.
feeling tired, seeming rested, looking good, sounding awful, tasting sweet are all gerund phrases used as stative or copular verbs or sense verbs. (This is a simplification.) They can all be made into sentences: I feel tired; He seemed rested; They looked good; You sound awful; It tastes sweet.
In their gerund form, they can be used as adverbs:
I went home feeling tired. [feeling tired describes in what condition, or how, I went home.]
She sat on the couch looking exhausted. [How she looked while sitting on the couch]
The child rode his bicycle down the street looking great. [how he looked riding his bicycle]
He left the house seeming rested but then realized he was tired later. [how he seemed leaving the house.]
Phrases that describe how some action is performed are adverbial phrases.
"I went home trying to sleep" is grammatical but does not make sense because it would mean you were trying to sleep while you went home.
"I went home and tried to sleep".
