This is more of a question for /r/asksocialscience but I'll tackle it from a historian's perspective.

Your answer to both questions is, yes. The Queen holds what are called reserve powers which give her the leeway to do a great many things. The last time a monarch used her reserve powers to affect Parliament in a way similar to what you are referring to is the bedchamber crisis of 1839, where Queen Victoria's support of Lord Melbourne was seen as rather uncouth by the rapidly democratizing Brits. Only a few years earlier, Victoria's uncle, King William IV, had dismissed a sitting Prime Minister on his own accord and appointed another of an opposing party. The resultant election was largely seen as a negative reaction to this, and William's appointment lost, forcing him to reinstate the man he had sacked.

A more recent example comes from the two big Commonwealth realms. In Canada and Australia, the Governor Generals serves as the Queen's representative. In theory, she continues to hold most of the same reserve powers in these two realms as she does in the United Kingdom, however the GGs are basically there to take up the ceremonial role that she now serves in the UK. You know, signing laws, throne speeches, etc. However, theoretically, through their embodiment as representatives of the monarch, they do have some powers should they choose to use them. In the case of Canada, this came about in the 1926 'King-Byng Affair' whereby the sitting Prime Minister, King, asked the GG, Byng, to dissolve Parliament and hold an election. Byng refused. You see, at this time, the Canadian Parliament was split between King's Liberals and the Conservative Party, but neither held an absolute majority. A third party, the Progressives (who would eventually go on to merge with the Conservatives) held sway. In fact, King's Liberals had fewer seats than the Conservatives did, but with Progressive support could maintain the confidence of the House. This had all come about in a mid-1925 election, and some in Canada were a bit miffed that despite having fewer seats than the Conservatives, King refused to resign as Prime Minister. Well, fast-forward a few months and he is asking for a new election based on a bribery scandal of one of his ministers that robbed him of his Progressive support and thus control of the House. Byng refused, as was his right, and offered the job to Arthur Meighen, the Conservative leader. Well, King's Liberals went from battling a scandal to being the suffers of perceived absolutist principles from the King's representative, and Meighen himself subsequently lost support of the house, thus forcing an election. Low and behold, the Liberals won big; Meighen even lost his own seat for even being associated as benefiting from Byng's choice.

Fast-forward to Australia, 1975. In this case, things were a little more complicated because Australia has an elected upper house. In 1975, the lower and upper house were controlled by opposing parties, and government was coming to a standstill. In Australia, the sitting government still has to control the lower house though, and Australian Labor Party lleader Gough Whitlam was the current PM. However, he controlled a razor thin majority in that lower House over the Liberal Party led by Malcolm Fraser. However, the Liberal Party had control over the Senate, and thus, could block legislation. Long story short, the sitting Governor General, John Kerr, saw that the impasse was negatively affecting the country, and through a few backdoor deals basically sacked the sitting PM and appointed Fraser as PM, who then quickly passed the blocked legislation in the upper house. An election was called shortly after, with funding for government secured. However, unlike in Canada, the scandals and poor governance of the sacked PM wasn't overcome by the actions of the Governor General, and Fraser was elected PM.

In both these cases, the GG utilized powers that theoretically remain to the Queen in the UK (as well as in her other realms, as the GGs in this case were simply acting in the Queen's stead; she could theoretically replace any one of them at any time and take the literal throne in any of these realms). Also, in both these cases, nothing was really done to resolve the issue. In both cases, a constitutional crisis emerged, particularly for Australia, however, in both cases, the actual issue wasn't really addressed. You see, the Queen's powers that are currently enacted by the government in Westminster are essentially at her pleasure. She continues to sign legislation, the act of which is required for any bill to become law. She continues to be the official leader of the military. Indeed, here is the current Oath new members of the military of the UK must swear: "I, (Insert full name), do swear that I will be faithful and bear true allegiance to Her Majesty Queen Elizabeth, her heirs and successors, according to law. So help me God." So the loyalty is to the Queen, not the state. At least theoretically.

Here's the rub. Despite having all these powers, if the Queen were to ever attempt to actually use them, it is very unlikely that it would have a favourable outcome for her (or the country). Speculation is discouraged on this forum for good reason, but I can't personally imagine how the Queen would be able to rule if she tried. The Brits have been more than willing to remove sitting monarchs in the past for disagreeable politics, lest we forget their invitation of Dutch invaders two centuries ago.