This is one of the questions on this sub I feel somewhat qualified to at least partially answer, but first I feel its necessary for me to correct some of the implications of it.

First things first, Pedro II growing tired of ruling Brazil does not necessarily mean that he didn't do a commendable job ruling, and the fact that the monarchy was ousted in a coup does not also mean that the regime was not popular. Now that I've established this, i'll answer.To answer your first inquiry, by the late 1880s Pedro II had become completely disillusioned with his role as Emperor and the role of the Monarchy within Brazil. This is likely due to the fact that Pedro had no heirs which he deemed suitable to replace him should he pass away and because of this he had eventually come to conclude that after his passing the Monarchy would not be the right institution to govern Brazil. Even despite having a daughter, Isabel who proved herself capable of ruling competently in his absence Pedro did not feel like she had what would have been required in order to lead a prosperous Brazil. Furthermore, Isabel was not particularly interested in creating spheres of influence within Brazilian elite circles in order to gain supporters and establish a legitimate claim to the throne either. Isabel did have children who may have been able to be heirs to the throne, however they weren't of the Brigantine dynasty. Knowing this, when Pedro's health began to decline throughout the 1880s he had taken it as a sign telling him that Brazil's time as a Monarchy was over.

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Furthermore, while this is slightly treading into speculation, at times it seems like Pedro was not particularly interested in ruling Brazil to begin with. In order to understand this perspective, we have to analyze Pedro's childhood. Pedro's mother, Maria Leopold died when he was only a year old. After this his father married Amélie of Leuchtenberg who Pedro had eventually begun to see as being a motherly figure in his life. Unfortunately his father, Pedro II whose influence in Brazil was waning as the country began to seem to be starting to fragment decided it was best to abdicate the throne and depart to Europe, taking Amélie with him. Adding to this, Pedro II had little friends his age, little contact with sisters who could serve as affectionate figures to him. While Pedro did indeed have guardians who acted in ways that allowed for him to have some sort of figure to look up to- at that point the damage was already done. As a result of all of these factors, Pedro had become a person who preferred the company of books and other things associated with a studious individual, and used them as a coping mechanism for his struggles. Now that we understand this important context, lets continue. Some would posit at this point that Pedro II had a dual identity, the former being a workaholic emperor who sought the best for his people, and the latter being a studious individual who preferred arts and sciences over governance, which he saw a burden. We see proof of this being the case as through Pedros letters wherein would claim he was born to devote himself to cultures and sciences.

Now maybe you understand why Pedro was reluctant to reinstate the monarchy. It had been a burden to him for much of his life as he preferred literature and science to the art of governing. Furthermore, the death of his sons made him embrace a fatalistic mindset that made him believe that the Monarchy was no longer the way for Brazil to go, and that when he is dead, it would be over.

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Next, regarding the coup. In my opinion we can usually separate coups into one of three camps. The former being a coup inspired by social/societal demand for change (As in the people want things to change and as a result of this opportunists take advantage of this and usurp the government, promising said change.) The second being a coup inspired solely by the desires of Elites within the highest echelons of society. The last being a coup inspired simply by instability and opportunism.

The coup that overthrew the Monarchy, was the second. a vast majority Brazilians had no interest in changing the way Brazil was, republicanism within Brazil was reactionary and Elitist, and as a result of this it was only popular within Elite circles. Even though they tried to compare themselves to the American and French revolutions, it had always failed. The reason why Elites within Brazilian society wanted to overthrow the monarchy was that during the 1880s, Pedro II had begun to take a less active affair in governance due to his health and instead acted as an observer. This made many elites believe that the monarchy was an unnecessary system that did little to no good for Brazil, especially because this was a new generation of elites who had never seen the transformation of the country from how it was under Pedro I to Pedro II, to them.

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Tl;dr Pedro II was someone who was more interested in books and science than governing, so by the end of his life he had simply become disillusioned and tired with governing especially because he didnt believe he had a suitable heir, so he was not interested in reinstating the monarchy, as he likely felt as if it were an unjust burden thrust upon him.

Tl;dr 2: Pedro II was overthrown in a coup because Brazilian elites had begun to see him as being no longer necessary for the success of Brazil. However its important for you to make the distinction and understand that just because someone is overthrown in a coup hatched by oligarchs and aristocrats does not mean they were not loved by the average person/the masses

Edit: I wanted to quickly go through some of the reasons for the Coup and give some thoughts. The abolition of slavery alienated landowning elites who formed one of the major factions that supported the Empire. Alienation from the catholic church and a disdain within the military for the existing order due to the fact that civilians usually outranked generals combined with increasing positivist influence led to the overthrow of the Monarchy

Hopefully this is a readable and acceptable answer for you, feel free to inquire more and ask for sources/recommendations.

Well, l feel like l can kinda of answer the first question.

See, for a time, when the royal family of Portugal ran away from Napoleon, they went to Brazil and eventualy made it a United Kingdom, which meant a lot more freedom than Brazil ever had as a colony, like being able to buy things from places others countries besides Portugal. Which didnt amount to much, since it meant that isntead of only trading with portugal, now its england.

Now, when D. João VI had to come back to Portugal, the country was in a bad place - to say the least - and one of the things Portugal wanted was Brazil back to being a colony.

So, Pedro (the first, but now shall only be named as Pedro) was the prince regent in Brazil and the people around him werent liking what was happening, and were pressuring him into doing something. Portugal had already sent an army, told Brazil to close the ports and sent an ultimatum to Pedro to come back to Portugal, so not exactly a good time.

Pedro, at this point, wasnt really on the side of Portugal anymore. The portuguese military had obligated Pedro to banish loyal allies - which was an humiliation for him - he was demoted to just "governor of Rio de Janeiro" when Portugal started taking hold of power as colonizer again, he was openly mocked by them and his wife, who agreed with the brazillian side, wanted him to stay.

Meanwhile, two groups were born the Liberals and the Bonifacians - the later one being led by José Bonifácio, who is a very important figure to the revolution, and if you want to read more about it, will pop up a lot - both with very different ideals of what should happen to Brazil, but agreeing that they didnt like the idea of going back to being a colony and wanted to Pedro to stay in Brazil as a way to defy Portugal.

So, Pedro decided to stay, in a day known as "dia do fico", before "dismissing" the Portuguese army and made José Bonifácio his minister of kingdom and foreign affairs, which made them closer allies. Tje liberals tried to stop this by making Pedro the "perpetual defender of Brazil" and convincing him that a contituent assembly should make the Brazillian constitution. A thing to note is that both sides had no problems with Pedro being king of Brazil, and most of their disputes involved who should make the constitution and their main ideals like what should happen about slavery, as an example.

Portugal, not taking very kindly to Pedro's decision, stripped him of all his powers, anulled Bonifacio's position and ordered Pedro to move that same instance to Portugal. When he learned about this from a letter in 7 of September, he was coming back to Rio de Janeiro from São Paulo, after a trip to secure the loyalty of the province. Legend says he unsheathed his sword and cried out "Independência ou morte" (independence or death), after ripping his blue and white armband that symbolized Portugal and asking his companions to do the same.

They returned to São Paulo, where they were received with celebrations from the public after the news and people, specially his companions, started claiming him "Brazil's Protector", "king of Brazil" and, at some point, the Liberals claimed him to be "Emperor of Brazil" and in the following days would start the suggestion of making him the constitucional emperor. Emperor was chosen because the word "king" brought with it the idea of absolutism and "emperor" brought the idea of ancient Rome or even Napoleon, somebody on power by popular acclamation.

Of course, Brazil wasnt really independent until Sepetember 22, but thats another story entirely.

Tl, dr: Brazil was a monarchy because most of the people who made the independence possible - prince Pedro, the liberals and José Bonifácio to name a few- were pro constitucional monarchy and were all working together for such and it was an "Empire" because the word King and kingdom brought the ideas of absolutism and the continuation of the Portuguese monarchy, which the word "emperor" didnt.

I hope that was helpful and please ask away if l left something out.

Btw, if you speak portuguese - or are lucky enough to find it in english - the book "1822" (and all the others in the trilogy, actually) by Laurentino Gomes is a great read about all that happened with the independence.