None of your examples have a word with an -ing at the beginning of the sentence. The phrases that start with -ing words aren't even clauses, at least using the traditional definition of a clause. What you're looking at are participial phrases.
A large crowd was still huddled around Bird, blocking him from sight.
This sentence includes one independent clause and one participial phrase. The clause has "a large crowd" as its subject and "was ... huddled" as its verb. The adverb "still" and the prepositional phrase "around Bird" modify this verb.
The participial phrase can be seen as a modifier of the subject of the main clause.
"He's going to be okay," Mrs. Banks interrupted, seeing the frightened looks on her son's faces.
This sentence contains two clauses. In the first, "he" is the subject, "[is] going" in the verb, and "to be okay" is the subject complement. In the second, "Mrs. Banks" is the subject and "interrupted" is the verb. The participial phrase at the end modifies the subject of the second, closer clause.
Your third example is quite similar to your second. The clauses "Mrs. Banks interrupted" and "Greg blurted out" function as dialogue tags.
Your fourth example demonstrates why we don't traditionally consider participial phrases to be clauses that need subjects.
Shari and Greg, running side by side at full speed, reached him together.
We have "Shari and Greg" as a subject. We have "running side by side", which does not contain a finite verb so it doesn't use "Shari and Greg" as a subject. Finally, we have "reached him together".
This "reached" is the simple past tense form. This is the verb that takes "Shari and Greg" as its subject. It's true that "reached" is also the way we spell the form that we call a past participle -- but here we know it must be the simple past because we have a subject that needs a finite verb.
This "running" represents optional information. This "reached" is the most important word in the clause. If we want the action of running to be the important action in the clause, we an write something like this:
Shari and Greg ran side by side at full speed, reaching him together.
If both "running" and "reaching" had their -ing forms and if there isn't some other verb with a simple present or simple past form, then there wouldn't be any verb that can use "Shari and Greg" as its subject.
Something has to form the predicate of a clause, and an -ing verb simply doesn't do that.
