To put your question in context, the 50+ year period of Dom Pedro II's rule (1831 to 1888) was the twilight of absolute monarchies in Europe. There were republican uprisings throughout western Europe and the German states in 1848-1850, and by 1880 even the militaristic new German Empire had a representative diet.

In the Empire of Brazil, the elected Assembleia Geral was closely modeled on the Parliament of Great Britain but wielded significantly more power. Throughout his reign, as the most important figure in Brazilian politics, Dom Pedro II worked with republican institutions in Brazil. The Emperor of Brazil had some veto power, but this was rare for Dom Pedro II to exercise. More typical, he manipulated events through force of personality and opportunity - most notably the abolition of slavery. Dom Pedro II was never an absolute monarch.

To answer your questions,

1 - his attitude originated in his education and sense of purpose which he privately remarked upon in correspondences (sources alongside deeper explanations). A lonely child, Dom Pedro II spent most of his youth buried in books and as he grew, modeled himself as an 'enlightened monarch'. Dom Pedro II was deeply intellectually curious, and as man of his times (and the last New World monarch after 1867) did believe that the people had a right to some representation. Nevertheless, he was not necessarily recorded as being explicitly pro-republican vs. anti-monarchy, rather being most concerned with what was best for Brazil - and that the Brazilian people should decide. That said, Dom Pedro II tacked closely to Brazilian public opinion. When Republic sentiment was high, Dom Pedro II was vocally more republican. There is also some evidence he was positively influenced by a visit to the United States in his youth, though I cannot find a reliable source for this outside a single article.

2 - there is not much historical record concerning other European monarch's reactions to Dom Pedro II's opinion of Brazilian republicanism. Most available correspondences consider how physically impressive and tired he was:

"He is very tall, broad, and stout, a fine looking man," writes Queen Victoria, "but vary gray, though only 44"

Contemporary royal correspondences capture an Emperor preoccupied with improving the scientific and industrial standing of Brazil. Dom Pedro II was keenly aware of his nation's backwardness versus the empires of Europe, and corresponded regularly with the leading scientists of the day. However, he did reveal his political feelings to Victor Hugo:

"I have no rights," he told the novelist, "I have nothing but a power resulting from chance; it is my duty to use it for good, for progress, for liberty."

Everyday Brazilians, however, were highly aware of Dom Pedro's republican inclinations. Dom Pedro II regularly and publicly defended the rights of Republicans in Brazil. There are many documented instances: Dom Pedro II condemning a mob attacking a newspaper, Dom Pedro I paying for the funeral of newspaper editors, Dom Pedro including prominent republicans in government cabinets. However, Dom Pedro also felt a centralized, powerful government was the best way to prevent Brazil from fracturing and improve the lot of everyday Brazilians: he publicly supported the Conservative party and as he aged, further meddled in politics, alienating him from segments of Brazilian society. Pointedly, he did not support the Republican Manifesto of 1870, but also refused to ban Republicans from government, instead declaring that if he was to be ousted from government, he'd become a teacher.

3 - Dom Pedro reacted with indifference and resignation, ascribing the coup to the will of the people. Furthermore, by his deposition in 1888, he was old (63) and sick. Dom Pedro II was eager to leave public life. How Brazilians responded to the coup depended largely on the social strata. Educated urban youth and landed rural elite were both frustrated by Dom Pedro's support for conservative causes near the end of his life and were eager to enhance republican power in Brazil - if for totally different reasons. Conversely, the poor and Afro-Brazilians associated Dom Pedro's monarchy with protection and freedom, and were devastated by his departure. Wikipedia cites an interesting passage in the linked source, in which Afro-Brazilians citizens tattooing themselves with the Imperial crown as a form of memorial, which is historically poignant considering tattoos were tools of ownership in slave societies. In all cases, his exile was extremely controversial - even among radical republican circles - and his legacy was almost immediately appropriated by all political sides: to Republicans, Dom Pedro II embodied the best of their values; to Monarchists, the stability and magnanimity of an enlightened king. And true to his character, the Emperor was buried with a book under his head.

Edit: clarified language around slavery.