1.Does this paragraph looks natural for a native to write?
Yes. There is nothing wrong with this sentence.
2.What does もの mean in 憧れていたもの? Is it talking about a physical thing?
Yes. もの can be translated exactly as a "physical thing".
3.In それは携帯電話ではなく, why is ではなくand adverb, why not ではない(verb)
According to "A Dictionary of Intermediate Japanese Grammar", page 211:
なく: a continuative form of ない used in written Japanese (to indicate a reason/cause for what follows if ない is attached to adjective) and contrast if it is attached to noun+copula.
Instead of なく you can also say なくて. Te-form is also used as a continuative form.
ではない can only be used at the end of the sentence or before a noun.
4.Does the bold part translate to: "At that time I was a student but, there was a thing I wanted, that thing wasn't a cellphone, it was a pocket bell."
I, who was a primary student at that time, was in love not with cell phones, but with beepers.