I think you're confusing multiple をs in a sentence with multiple をs in a clause.
Multiple をs in a sentence is perfectly normal:
whereas multiple をs in a clause isn't:
What makes your example sentence complicated is that it's not completely clear whether there is one clause or two.
In English, the word "and" is very flexible, and you can use it to combine almost anything from clauses and verb phrases to noun phrases, adverbs and adjectives.
In Japanese, you use 連体形 (~て、~で) to combine clauses and verb/adjectival phrases, と or や to combine noun phrases.
In the example sentence, what is being combined isn't just (object) noun phrases (ギター and 作曲) but also prepositional phrases "hanging off" them (14歳の頃に and 17歳で). In English, "and" works fine for this:
He begins guitar-playing around 14 and composition at 17
In Japanese, if it were just the object noun phrases, と or や would be possible
but with the prepositional phrases, this becomes unnatural
So the example sentence is "avoiding the problem" by simply listing the two object+prepositional-phrase compounds next to each other
Whether you choose to interpret this as two clauses where the verb is dropped from the first one, or one clause with two object+prepositional-phrase compounds hanging off the main verb... I don't see any reason why one is preferable to the other, but others might shed more light on that.
Is this substandard/casual Japanese more appropriate to mixi than a biographical snippet?
On the contrary, I would say that it's slightly formal/written/newspaper-language-like. In speech, I think people would tend to say