In 1800, about one-half of the Irish population still used Irish as their daily language, mostly those west of an imaginary line from Wexford in the SE to Derry in the NW, and an estimated one million were unable to speak English at all (or not more than a small amount). So, at that time your family in Tipperary was almost certainly still using Irish on a daily basis (especially at home), though the adults (especially the men) could probably speak some English, and in the larger towns English would have been more widespread. It's not that no one east of that line was able to speak Irish, but those areas were already well on the way to English becoming the norm. Pockets of Irish speakers continued to exist in the east, even some in rural areas near Dublin, through the 19th century.
Irish was not always a Catholic thing in those days, either. Protestant merchants, ministers, and officials were often able to speak it, and most anyone living in a heavily Irish-speaking area would generally be able to speak Irish, at least if they engaged with Irish-speakers, and for some it was even their native language. In fact, when the Irish Parliament voted itself out of existence in 1801 (forming the full UK as a result), some of the Members of Parliament from more rural areas (all of whom were Protestants) had to have the effects of the Act of Union explained to them in Irish, because their English was not good enough to understand it fully.
Far fewer people could read or write Irish, though, because it was not taught in many places after the Battle of Kinsale (1601), when the old Gaelic social and legal system collapsed, and its use was actively discouraged. Some people insisted on using written Irish, even in the courts and when dealing with the government, but over time the authorities put an end to almost all official use. So, if a record from the 19th century says that someone was literate, it probably means the person was able to read and write English, not Irish. Written Irish picked up again towards the end of the 19th century, with the Gaelic Revival.
Irish usage declined dramatically and surprisingly quickly as the 19th century went on, partly due to the schools (as noted above), partly with the collaboration of the Catholic church (currying favor with the authorities, especially after Catholic emancipation in the 1830's), and also greatly due to the uptick in emigration caused by the famine. People realized that English would be important in North America or Australia, and even in Ireland if they wanted to get ahead, so families gave in and cooperated with the schools to make sure that their children learned to speak English fluently. Even the great liberator, Daniel O'Connell, who was a native Irish speaker and gave many speeches in Irish in the 1820's and 1830's, nonetheless openly encouraged people to switch to English in order for Ireland to advance. He may not have meant that they should give up their Irish entirely, but that was the ultimate result in most cases.
I'm attaching a map showing that, as late as 1871, the area around Cashel still had a significant number of Irish speakers, though more northern parts of the county no longer did.
As noted above, anglicization of names began in the late 1600's, but it usually happened only when there was interaction with the authorities or a landlord's agent, someone who needed to keep a record of the name, so the records were kept locally and there was no standardization. It was a very inconsistent process, with names being spelled "phonetically" by the official or agent, or in some cases "translated" by someone with a weak grasp of Irish, or of the meaning or history of the name. There's an egregious example from your own family. The Irish form of Quirk/Quirke is Ó Cuirc. Its origin is uncertain, though it may come from the word corc, which is one of the words used for the heart (croí being more common nowadays). Most Gaelic surnames are based on the given name or nickname for a real or legendary ancestor, and the nicknames could appear insulting to us nowadays (Campbell, for example, means "crooked mouth", and Cameron means "crooked nose"). Maybe the original Curc/Corc (Cuirc being the genitive form = "of Curc/Corc") was a man who was considered to have a big heart. It also had anglicized forms such as Kirk and Quick, but probably the strangest one is Oates, which arose because someone "translated" the name, assuming that it was based on the word coirce, which means "oats".
Over time, anglicized forms could change, even in the same family and sometimes in the same generation. In one branch of my family, in the mid-19th century, one form of the surname was used for the earlier children, and then another form for the later children (possibly there was a new priest who caused the change). In many cases, the people whose names were being written down may have been unaware of what they were being called, until a later generation when they could read and write. Even then, they may have had little control over it, or been all that concerned, and among themselves may have gone on using the (spoken) Irish forms. That was also true of given names, especially when the priests started using "Latinized" forms or what they thought were biblical equivalents. So, a child might go on being Conchobhar (Conor) to his family, although the priest insisted on writing his name down as Cornelius. Some of the priestly forms later took hold, but some never did. A boy named Donncha may have been called Dionysius by the priest, but I've never heard of a case where that name stuck (Denis became a more favored anglicization). People who largely ignored what was written down as their name could easily choose a different anglicization later on. So, a Sorcha who was called Sarah by the priest in the parish register may, on emigrating to America, have chosen to call herself Sally instead. In the first half of the 19th century, people had no birth certificates or ID's to carry with them when they emigrated, so they were free to spell their names however they liked.
