In Dante's "Inferno," Tarpeia is not featured as a character. She is a figure from Roman mythology, known for her betrayal of Rome. In Dante's work, characters are primarily drawn from Christian theology and classical literature, rather than Roman myths like Tarpeia.

She was the daughter of the commander of the Roman citadel who tried to sell Rome out to a conquering enemy. All she asked for in return was what the soldiers wore on their left wrist, meaning their bejeweled gauntlets. They interpreted this as her wanting their shields so they threw them on her and she was crushed to death. They threw her body off a cliff which became known as the Tarpeian rock where Rome would later cast the bodies of only the most treacherous of people to sea.

No, this form of capital punishment did not exist.

The most gruesome forms of capital punishment were scourging (being beaten to death) ad bestia (devoured by animals) which was a sentence to the arena where you were left at the mercy of predator animals, and poena cullei (penalty of the sack). The person was sown into a leather sack together with a dog, a monkey and a cockerel. This was the worse penalty because it stripped dignity more than the others and because you would not be given proper burial.

Other forms of capital punishment were; beheading (percussio securi), this was the most common form, burning at the stake (this was for arson, slaves who conspired against their masters, deserters and enemies of the state) strangling in prison (strangulatio) throwing a person from the part of the prison called Robur (precipitatio de robore) throwing a person from the Tarpeian rock (dejectio e rupe Tarpeia) and throwing a person into the river (projectio in profluentem). For slaves and non-Romans there was also Crucifixion (in crucem actio). Roman citizens could not be crucified.

Most people who were sentenced to death, especially for the most gruesome forms of punishment, were poor people. The rich mostly found ways of getting out of it or were treated leniently, except for cases of parricide, where even the most gruesome forms of death were not spared.

The Vestal virgins (the priestesses of the goddess Vesta) who were caught breaking their vow of chastity were buried alive.

There was also free decision of death (liberum mortis arbitrium) where you were allowed to choose the method of execution. This was in essence an order to commit suicide, which was preferable to other sentences because it spared the humiliation of a public execution, your property was not confiscated, and you would be given a proper burial.

Anything. Being a patriarchal society, Roman men had absolute power over their children, wives, and sometimes even extended family in certain cases. They could kill them if they liked, though that was looked down on. So were many harsher punishments; you must remember that these children (the boys, at least,) would grow into the next tier of Roman aristocracy in many cases. They weren't mistreated too often or severely.

Striking a child would be a simple and common punishment. We don't have exact records on Roman society, I'm getting this last punishment from a passage regarding a group of rowdy children refusing to do their schoolwork. Most of our ideas on Rome are from reliefs and writings, and people don't often carve pictures of them beating up kids. Unless you're Caligula, in which case graffiti might depict something similar. But I digress.

I hope I answered your question.