The sentence is fine as it is. It's not a run-on sentence.
The answer you reference is discussing how it's ungrammatical to join two independent clauses with a comma and without a conjunction. But the sentence in your question is a main (independent) clause that's followed by a nonessential relative (dependent) clause.
Consider your sentence:
The musical scene was a maze of competing movements and slogans, many of them claiming to be "new" in one way or another.
You could drop everything in italics and still have a grammatical sentence. Syntactically speaking, what follows the comma is simply acting to give additional (but ultimately unneeded) information about the main clause.
Other examples of this would be:
I walked carefully up the stairs, being careful to mind my sore foot.
I walked down the hallway, cautiously avoiding its freshly painted walls.
I joined my friend at the bar, who had started drinking without me.
Past participle verbs will work with other verbs of any tense, so claiming will always work (although you may need a helper verb along with it—such as were claiming, are claming or will be claiming—in some constructions).
It doesn't have to be claiming, but it can be. If you use something other than just claiming, you need to make sure it uses a verb tense that matches what's used at the start of the sentence.
Take your first alternative sentence. It doesn't use claiming but it's a fine alternative because it uses the past tense:
The musical scene was a maze of competing movements and slogans, and many of them claimed to be "new" in one way or another.
We could also take your second alternative sentence (which, unlike the first, is another example of the use of a relative clause) and change both verbs to the present tense:
The musical scene is a maze of competing movements and slogans, many of which claim to be "new" in one way or another.