The existing answers provide an excellent background to the political situation in the nineteenth century. I will try to answer the specific points raised in the question.
Before I begin, I would like to clarify a few points.
Firstly, the ability of the monarch to dissolve Parliament comes under what is known as the Royal Prerogative. Since the "Glorious Revolution" and, in particular, the 1689 Bill of Rights that followed it, the exercise of Royal Prerogative has been limited. Article 1 of the Bill of Rights states that the:
"power of suspending the laws or the execution of laws by regal
authority without consent of Parliament is illegal."
Further, the Bill of rights confirmed that Parliament had the right to limit the use of remaining prerogatives (which they did in the Triennial Act of 1694).
In practice, this meant that the monarch could no longer dissolve Parliament without the consent of Parliament. (There was a specific exception to this however. Parliament was dissolved on the death of the monarch, as happened in 1820 for example, although - to be fair - this is very much an extreme case!)
The second important point is that, in the nineteenth century, the monarch appointed the Prime Minister and had the absolute right to appoint whomsoever they wished. Obviously, this could - and did- cause problems. In fact, this had been the cause of what is now known as "the Decade of Ministerial Instability" under George II in the previous century. HM government have an interesting article of the development of the institution of Prime Minister on their website.
Finally, we should remember that political parties in the UK only began to coalesce into the kind of parties that we would recognise today in the late eighteenth / early nineteenth centuries, over the period from about 1760 to 1834. Previously, the "Whig" and "Tory" groups in Parliament are best thought of a loose coalitions of MPs with broadly similar views and goals. However, there was no "party line" on particular bills that came before Parliament. By 1834, the groups had become sufficiently well established that Robert Peel could issue the Tamworth Manifesto defining the goals of a "Conservative Party".
The split was effectively complete by the watershed election of 1852, where the two-party system of Conservative and Liberal parties emerged.
So, to answer your specific questions:
Under what circumstances were 19th Century parliaments dissolved, and what motivated the principle actors?
Normally the Prime Minister asked the monarch for a dissolution of Parliament. This could be because they could not command the confidence of Parliament or form a stable government (e.g. in the election of 1807).
Before the emergence of the two-party system in the middle of the century, contentious legislation on issues like Catholic emancipation or Parliamentary reform simply caused the various political coalitions to shift and reform. A Prime Minister on the "wrong" side of such legislation could easily lose the confidence of Parliament (or of the monarch who had appointed him).
Contentious legislation after the new parties had begun to emerge in the 1830s, such as that for the repeal of the Corn Laws, could split the newly formed parties and either strengthen the position of the Prime Minister (as was the case with Robert Peel in the election of 1841), or fatally undermine it (Robert Peel resigned in 1847, rather than asking for a dissolution of Parliament, fearing that the forthcoming election would become a vote of confidence).
Prime Ministers could also ask for a dissolution of Parliament because they sought to obtain a political advantage. This was the case in the election of 1806. Then, as now, such attempts to gain political advantage at an election were not always successful. Another general election followed in 1807!
As mentioned above, a dissolution would be triggered by the death of the monarch (e.g. the election of 1820).
In addition, in the eighteenth and nineteenth centuries it was normal for Prime Ministers to call for a dissolution of Parliament following an Act of Parliament that resulted in significant changes to the electoral system. This happened, for example, in the 1832 election following the Reform Act of 1832. However, when such an act came late in the life of the Parliament, as with the Representation of the People Act 1884, the election might be delayed, as happened with (the 1885 election).
Between the Prime Minister, Parliament, monarch, electorate, etc., who was able to exert influence to force a change?
The Prime Minister could ask the monarch for a dissolution of Parliament. Then, as now, meetings between the monarch and their Prime Minister are private, so we do not know how often, if ever, the Prime Minister's request was refused.
Parliament was able to make the Prime Minister's position untenable. In such cases the Prime Minister could either resign or ask the monarch to dissolve Parliament and force an election.
By the nineteenth century, the monarch no longer had the authority to dissolve Parliament unless asked to do so by Parliament itself, normally in the person of the Prime Minister.
The electorate had no say in the matter.
Click on the "election" link in the Wikipedia list of UK Parliaments for more details about each election.