Does the Queen have any say in what she does or who she has to meet or dine with? What would happen if she stamped her foot and said I won't do that?
Is there anyone who could force her to do something?
Yes,, I totally agree, and what she does, whether she wants to or not, is done for the good of Britain, first and foremost, and while there have been some hiccups along the way, that is what the Royal family stands for, the greater good of Britain.
...and if she was really adamant about not responding to a particular invitation or situation surely she does possess the Royal prerogative of proclaiming "We are not amused; therefore, off with their heads!"
thinking about it rationally i think the following
1) she DOES have a choice
2) She probably never exercises that choice because she grew up knowing she would have and does have a "duty" to her country, much more so than other people.
3) if she did decide to exercise her choice, i can't see how anyone could physically force her to attend a function/trip against her will