Can we say that Latin Hexameter did not exist before Ennius (as implied later in this article)? Not necessarily written Hexameter is in question.
We can't say for sure, but Ennius' hexameters are the first extant Latin hexameters. If it existed prior to him, we do not have an example of it, and his immediate predecessors (Andronicus and Naevius) wrote in Saturnian meter.
When were the musical properties of Greek Hexameter "forgotten"?
They weren't forgotten, per se, they just weren't natural. And the shift from oral epics to literary epics happened before Ennius. There were probably still traveling bards, but the major literary works transmitted to us from the 5th century are all working in a literate (or semi-literate) society.
If almost nothing was left from the Greek Hexameter but its "rules", then on what account it should have been adapted by the Romans(*)? and all this just for developing, at the end, "characteristics of its own" and align with "natural result of musical expression".
I'm not going to get into defending Wikipedia (it's just Wikipedia after all!), but I don't think they're saying what happened was bad, just that it happened. There is a big difference between Homeric oral epic and later literary epic. The latter was recited, of course, but it was an external system built for one language that was artificially adapted to another, displacing the earlier, native system (Saturnians).
This is what the article is talking about when it discusses spondees and dactyls. The most common configuration of feet in Homer is four dactyls (followed of course as they all do by a dactyl and a spondee at the end). By contrast, Ennius' favorite pattern was four spondees. In general, Latin hexameters feature more spondees than Greek hexameters.
If you want a breakdown on this with more statistics (which is far more complex than the narrative given above), you can read Duckworth's immense 1966 study "Studies in Latin Hexameter Poetry" (TAPA 97: 67–113).
Another area concerns oral aspects. Epic poetry of the Homeric variety was composed for memory and extemporaneous recitation. Ennius and thereafter (and the Hellenistic counterparts like Apollodorus) on the other hand more favored poetics. This isn't to say Homer lacked them, but the repetitive lines and epithets which were chiefly used to aid in recitation were replaced with more meaningful (to the later Greeks and Romans) expressions.
(I personally would argue that certain epithets were sometimes chosen to fit the narrative, but clearly more so in later epic than in Homer).