Well I THINK they thought they found the three crosses way back when and sent out pieces of them. But there is zero proof that it is the real cross. I think some queen back in the day thought she found them.
Highly doubtful. It as made of wood and its been 2000 years! Besides crosses were a dime a dozen then. There were other criminals crucified at the same time as Christ. Most likely once he was taken down the cross he died on was used for other executions so how would anyone know which was his after awhile? I doubt his mom or other followers would have been too keen in dragging two huge, heavy beams of wood about even if the Romans -who owned the cross - had let them. Its only we modern Christians who are so obsessed with an object of brutal execution and suffering. Its a good thing they didnt have other methods then or we'd all be obsessed over a hangmans noose or an electric chair now.