Do indulgences and inquisition come under papal infallibility. Why or why not?

The doctrine of infallibility was dogmatically defined by the First Vatican Council in the year 1870.  Even though more than one hundred and fifty years have passed since then, many, both within the Church and outside, have found it difficult to understand what it actually means. To some it means that the pope is impeccable, while to others it is some kind of divine inspiration. So what exactly is infallibility? 

“It is a divinely revealed dogma that the Roman Pontiff, when he speaks ex cathedra–that is, when, acting in the office of shepherd and teacher of all Christians, he defines by virtue of his supreme apostolic authority, doctrine, concerning faith or morals, to be held by the universal Church–possesses through the divine assistance promised to him in the person of St. Peter the infallibility with which the divine Redeemer willed his Church to be endowed in defining doctrines concerning faith or morals, and that definitions of the Roman Pontiff are therefore irreformable because of their nature, and not because of the agreement of the Church.”

Cardinal Newman, in his letter to Gladstone so often quoted says, “A Pope is not infallible in his laws, nor in his commands, nor in his acts of state, nor in his administration, nor in his public policy.” He is liable to err in mere matters of opinion like other men. The supreme civil judge may write a letter, or give an opinion by word of mouth on some point of law, but it would have no binding force, whereas the same opinion delivered ex cathedra — from the “bench”—would be binding and final.

In short, infallibility simply guarantees that the official teachings of the united Catholic episcopate and the definitions of the pope regarding Catholic doctrines are free from error. 

Coming back to the question, let us now look into both indulgence and inquisition and their relationship to papal infallibility. While indulgence is a Catholic doctrine, the term inquisition refers to a special ecclesiastical institution for combating or suppressing heresy. As the reader can see clearly, indulgence comes directly under papal infallibility as it is a doctrine of faith and has been officially defined by councils like the Council of Trent. On the other hand the inquisition is indirectly linked to papal infallibility since they make use of the official teachings of the Church to combat heresy. 
Having said this, it is only right to point out that though indulgence and inquisition are not related to papal infallibility in the same manner,  both these terms have been gravely misunderstood and misrepresented. Yes it is true that both these institutions were abused by some men, but if we are to look at  reliable data, instead of prejudiced and false history, we would find that the Catholic Church is not really responsible.