2. Gerunds and participles are two grammatically distinct things that have no overlap as the "being" examples illustrates. Note that "participle" has two usages: 1) the verb forms present participle (being, going) and past participle (born, gone), and 2) a "participle clause or phrase". Participle clause is a very useful and versatile way to construct an adverbial clause.
Using your example sentence,
* The children playing in the park are screaming loudly.
Here "playing" is a present participle (modifying "the children") in the noun clause "the children (that are) playing in the park", and the noun clause serves as the subject of the sentence. Participles often creates an adjectival phrase like this usually coming after the noun they modify (e.g. "people suffering from pneumonia", "infants born last year"). But these phrases are NOT participle clauses.
* Playing in the park, the children were screaming loudly.
"playing in the park" is a participle phrase. The key difference is the clause is (usually but not always) separated from the main one, and it can take the subject from the main one (although it can sometimes have its own separate subject too - there are "same subject" and "different subject" participle clauses), and finally it has a simplified tense expressions.
In this example, the clause is separated by a comma, and takes the subject "the children" from the main clause, and indicates the same tense as the main one.
Tense can be the same ("same tense" participle clause - "playing") or back-shifted ("perfected participle" clause - "having played").
A participle clause/phrase is particularly useful in that it is shorter because it does without a conjunction (no "as", "while", etc) and has a simplified tense expression. They "participate" in the sentence by adding exactly what the speaker needs to express.