Is it true that Apostle St Peter was crucified on a cross and he ask to be crucified upside down?

Yes, just visit the "Domine Quo Vadis Chapel, Rome"

The Chiesa di Santa Maria in Palmis, better known as Chiesa del Domine Quo Vadis (literally, the Church of "Lord, Where Are You Going?") is a small church on the Appian Way in Rome.

History of the Domine Quo Vadis Chapel

The Chiesa del Domine Quo Vadis is located on the spot where tradition says Saint Peter had a vision of the risen Christ while fleeing persecution in Rome. According to the apocryphal Acts of Peter, Peter was surprised to see Jesus and asked him:

that is true and some others' also followed Peters' example.

Yes - just outside Rome.

Yup! That's correct.

Well...

the only information we have on the matter of Peter's death is what you have described.

We do not have enough information to tell for sure if it is the truth or if it is just a legend.

It's possible I guess. But the earliest writings to it are from Origen & Tertullian and neither were born yet when it supposedly happened. They weren't born until the middle/late 2nd century.

Yes, Roman executioners are known for their kindness and consideration for the people they execute and would always allow their requests.

There's no Biblical evidence of it, but that's the story that's been told for ages.

That's the legend.

However, most of those hagiographical stories contain a great many things that are entirely undocumented and unverifiable.

Could be possible but could also be a rumor. There is no official writing on the matter.