Excellent question. First, let me give you my somewhat rollercoaster answer: while I would attribute the bringing about of the Ausgleich to political dynamics beyond the scope of individual dynastic personalities, Elisabeth of Bavaria definitely played a massive role in the symbolic and real union of Austria and Hungary in 1867; however, scholars do agree that her role as reported by popular history has been somewhat inflated.

So as you can see, I've said, "no, Sissi didn't play a role," and then, "well, yes she did," and finally, "but take that with a grain of salt." Let me unpack that a bit for you.

You're right about Sissi's political role as Empress of Austria. As per Alice Freifeld:

Elisabeth, as the wife of the Emperor Franz Joseph, appeared on no bureaucratic organizational flowchart, but she had direct access to the Emperor for her causes. Like other royal consorts, she did not herself possess the power to impose policy or govern, but she could serve as a conduit.

Sissi's power was soft power, the ability to control to some degree the issues that Franz Joseph saw on a given day, and in many cases, the people who came to present those causes. You're right that the Hungarians themselves "pushed for the status." The failed revolution of 1848 (quelled only by the threat of Russian troops on Hungarian soil), the subsequent liberal-constitutional experiments of the 1850s, and the Austrian defeat in 1866 all pushed Franz Joseph towards accepting the Ausgleich in 1867. So no, Elisabeth of Bavaria did not directly create the policy of the Ausgleich, nor the political conditions whence it came.

However, Sissi did a great deal to make the Ausgleich palatable to both Vienna's court circles and the Hungarian people. Elisabeth's love affair with Hungarian culture is well documented. I've not read Reluctant Empress, but I can't help but assume that Hamann mentions Elisabeth's learning Magyar and introducing its use to the fashionable circles of Vienna's power players. Again as per Freifeld:

[Elisabeth] surprised the court in 1863 by taking up the study of Hungarian, and amazed those at court even more by actually learning it. She helped stampede Hungarian Habsburg bureaucrats and society into learning the language and engaging its culture, reversing a decade of Germanization.

Through her ability to speak Magyar and her pbvious affection for the culture, she was able to cultivate important relationships with Hungarian movers, most famously Count Andrassy. After the loss to the Prussians in 1866, Franz Joseph dispatched Elisabeth to Hungary to serve as a vanguard of rapprochement and reunion. In Hungary, Elisabeth "played a critical role in bringing about the reconciliation between Hungary and the Monarchy." Once the decision for Ausgleich had been reached, the cult of Elisabeth as "savior of Hungary" began immediately.

As you may well know, the Ausgleich declared that the separate nations of Austria and Hungary would share a common Monarch and Dynasty: Franz Joseph and the Habsburgs. Sissi was absolutely vital to selling this concept of the Habsburgs as Hungarian monarchs to the Hungarian public. Again:

During the year and a half between her visit to Pest in early 1866 and coronation day [the day she and Franz Joseph were to be crowned Apostolic Queen and King of Hungary], 8 June 1867, Elisabeth was central to the pageantry of Compromise. Through her shone the new, human face of Habsburg royalty. Citizens clamored for a glimpse of her... No day seemed to pass without some new article on Elisabeth.. In so far as dualism was acted out as a marriage, the Hungarians played the role of the wife, and Elisabeth represented Hungary's role in the unique personal union which was established.

Basically, Sissi became a celebrity monarchy in Hungary, and the people of Hungary adored her. She was beautiful, skilled at equestrian, a tastemaker in fashion and (most importantly) spoke nearly flawless and spontaneous Hungarian (Magyar). In essence, she took the image of the Habsburgs as occupying conquerors that had existed in Hungary for decades, and transformed it into an image of the Habsburgs as "our Monarch" for the Hungarian people. The sheer media frenzy surrounding Elisabeth and the 1867 Coronation cannot be understated. Celebratory dishware, pins, postcards, paintings, lithographs, etc. overflowed in the streets of Buda around the time of the Coronation. And again, her decision to pick up Hungarian and master it did wonders to win over Hungarian politicians at court.

I have to go meet with some faculty on campus at the moment, but I'd be happy to finish up my response when I return, as well as hopefully address your subsequent questions.

In the meantime, however, your interest in Sissi allows me to wholeheartedly recommend a book and a specific essay.

Freifeld, Alice. "Empress Elisabeth as Hungarian Queen: The Uses of Celebrity Monarchism" in The Limits of Loyalty: Imperial Symbolism, Popular Allegiances, and State Patriotism in the Late Habsburg Monarchy. Edited by Laurence Cole and Daniel Unowsky (2009).

This essay is the basis of much of my answer. It also contains a great deal of information about Sissi's subsequent portrayal in popular media, both at the time of her life and death, as well as today. It also provides a good 'reading list' of fictional material based on Sissi's life, if you're interested in film, television, and theater. I can also recommend the entire book, as much of it deals with the role of monarchy and dynasty in the Austrian Empire.

Hope to get back and answer your other questions later, hope I could help for now.