I wrote an answer to a similar question about manumission in wills some time ago so I’ll transplant it here- TL;DR: there really isn’t any verifiable answers to your questions. It’s all guess work. The Romans didn’t/we don’t have any records that lay out the kind of statistics you are looking for. We don’t know how many slaves there were in any given place in the empire; we don’t know their total population across the empire at any given time; we don’t know the percentage that worked manual labour position compared to domestic ones. All we can say, based on the evidence, is that the vast majority lived and worked labour based jobs and they died poor and generally without recognition in hard conditions. Our typically snobbish upper class sources saw manual labour slaves (and slaves more generally) as all property, disposable, untrustworthy, lazy, lowlifes, uncultured, dirty and unwashed.

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Many owners would free slaves in their wills and these are widely used as pieces of evidence for historians to make guesses on the slave population in the Roman Empire; which may have been around 1/3rd the entire population of the empire at its height. But the law Augustus introduced in 2 BCE limiting the number of slaves that an owner could give freedom to in a will has more to do with the displays of wealth than resolving any kind of labor shortage or supply. It must be remembered that Augustus held the powers of censor, an office that was designed for the purpose of limiting opulence and extravagance among the senatorial class and thus preserving traditional Roman values (ala Cato the Censor) and downplaying the ability for certain senators or patricians from showcasing more wealth then their comrades. Augustus wished to present himself as the defender of traditional Roman values (in contrast to Mark Antony and Augustus’ own earlier career). He tried to downplay aristocratic rivalries that had shatter the Roman Republic in the various civil wars before his sole rule, he was always keen to instigate laws and decrees to achieve this purpose. The freeing of slaves in a will is a very clear statement of one's wealth. Imagine it, if an owner could free hundreds if not thousands of people upon his death (and not ruin his family or interests) than it was an extreme display of privilege and this could start up the competition. I would also like to point out that those who were freed seem to be those who were known to, connected to, directly served, or interacted with their master in some kind of capacity; usually in the household and at a business. The larger cluster of slaves on farming estates, in mines, or chained to the decks of merchant ships probably never meet nor saw their masters and therefore languish in horrible conditions for their entire lives. The point is, for every slave that was freed by their master and maybe went on to some prominence in a random town in the empire, a thousand died nameless in the worst and most dreadful conditions one can imagine.

A point can be made with the Augustan law referenced above. It designates different types of slave owners, those who own one or two slaves, five or ten, ten and thirty, and so on. This highlights that their was all kinds of slave owners and that they were available to even those in the lowest socioeconomic status, not just on the large farming estates with their chain-gangs worked. A man who owned one or two slaves likely could not or would not free them on his death because of the resources he had invested in them; maybe the slave was a secretary or accountant at a business, a maid or wet-nurse at his home, a tutor for his children, a laborer at his construction firm- the list could go on and on and is hard to determine definitively how often they were freed or what professions they occupied for the lower orders since we simply do not have the literary evidence from the non-aristocratic writers or from any slaves themselves.

Let me leave you with this virtual eye-witness account by Diodorus Siculus, where he describes the lives of slaves trapped working in Roman mines:

"But to continue with the mines, the slaves who are engaged in the working of them produce for their masters revenues in sums defying belief, but they themselves wear out their bodies both by day and by night in the diggings under the earth, dying in large numbers because of the exceptional hardships they endure. For no respite or pause is granted them in their labors, but compelled beneath blows of the overseers to endure the severity of their plight, they throw away their lives in this wretched manner, although certain of them who can endure it, by virtue of their bodily strength and their persevering souls, suffer such hardships over a long period; indeed death in their eyes is more to be desired than life, because of the magnitude of the hardships they must bear."