When a church has a relic of the True Cross it is a tiny sliver. All the slivers in Europe do not make a whole beam of the Cross. It is recorded history that St Helena, Constantine's mother, brought it back from the Holy Land circa 330AD.
Yes and there is no way of knowing if each sliver is from the original. Also Helena excavated the cross 300 years after the crucifixion. I don’t know if there had been other crucifixions there in the remaining 300 years?
The cross recovered by Helena seems to be the “most authentic”. However, early Christians did not use a cross symbol (they used a fish) because they saw the method of Christ’s death as shameful. Therefore I am not certain they would have endeavored to preserve the cross.
In any event, the sale of relics became a problem and many fakes abound, no doubt of slivers of the cross as well as saints bones, etc.
Was the original crown of thorns preserved? unclear.