It is conventional wisdom (though that does not make it automatically wrong) that children of the wealthy in ancient Rome were tutored by educated Greek slaves.
It makes sense from both the slave's and owner's incentives for the smart and hard-working slave to learn some valuable skill, at the least so he is worth more and has a slightly less difficult life. Maybe this includes some limited literacy, but music? literature? poetry? rhetoric? Even the finer points of grammar? That does not fit.
Where did these slaves who could teach high culture come from? Where are the incentives to give a slave that sort of non-marketable-skill education? How did a large number of Greeks with the trappings of being from the upper class fall into slavery?
Also, if the teacher was a slave and the kid being tutored was free and from a 'good' family, how did discipline work? I'm thinking every culture, on every continent and every century, up until the last ~2 decades, kept to the tradition that 'an idle fool is whipped at school.'
There were a lot of wars between the many small nations back then, as well as rampant piracy, a situation which constantly produced slaves. Also, debtors could become slaves at many points in classical history. This, along with enslaved/exiled political dissidents, would have been the source for "educated" slaves, while these means were supplemented by the acquisition of barbarian slaves for more menial tasks, such as mining. Children of slaves, obviously, were another source.
It made sense to educate a slave if they were to manage other staff, perform secretarial duties, etc., but often someone who was fairly literate already and had somehow fallen into slavery would be used as a tutor. An example is Livius Andronicus, who was a Greek poet enslaved when his city was captured in 272, tutored his owner's son, and was freed when that was completed. Many slaves were either freed in this way or bought their own freedom after acquiring money or property (yes, they could do that in Rome despite being under the dominium of a citizen).
Cicero helped to educate his secretary Tiro, who was a slave; he was manumitted and continued to serve Cicero, publishing his former master's letters and works after his death. This was an exceptional case, but likely he was not alone in receiving education as a Roman slave.
I can say nothing as to your last question, although I think it likely that in some cases, tutors were allowed to discipline children whether they were slaves or not.