Did ALL fathers in Ancient Rome have the right to kill their children, or only the "paterfamilias"?

Factual Questions
1

As a classics scholar, I learned that the “paterfamilias” or oldest male of a family had absolute authority over his children (including adult children). This included the power to sell them into slavery or even to put them to death. This power was eventually softened by the law, but it took a very long time, as in centuries. The paterfamilias, in fact, was the only member of the family who actually had the full rights of a citizen (this is why, at the time of the Catilinian Conspiracy, some of the conspirators planned to kill their fathers - so they would become full citizens according to Roman law and be able to exercise autonomously the powers that they wanted to gain through the conspiracy). A paterfamilias could emancipate his son, giving him full legal capacity, but that was akin to disinheriting him or severing him from his family.

What I am not clear about is, did Roman law give fathers who were not paterfamilias similar powers of life and death, the right to sell them, etc. over their children? Could all adults be disciplined and restrained by their fathers, regardless of whether or not he was the oldest male in the family? Or are the sources silent on this point?

Are there any known cases in Roman history where a paterfamilias actually used his power and had a member of his family put to death?

2

AFAICT, not without extra justifications and/or repercussions. See Ch. 2, Sect. 2 of this thesis.

3

Wiki seems to say not, but I haven’t followed up the sources:

Adult filii remained under the authority of their pater and could not themselves acquire the rights of a pater familias while he lived

Note there was an exception introduced under Augustus (the Castrense peculium), that whatever a son acquired in military service was directly his property that he exercised sole power over (but came to his father if he died). This was later extended to property acquired in civil service as well. But that only refers to property (including slaves, I would suppose). But one’s children wouldn’t fall under that. They are explicitly under the pater familias