Based on the evidence thus far, it does not appear that beliefs are something we arrive at by choice. Although we don't seem able to command our own beliefs at will, for some reason we seem to think that others can do this. We — and by that I mean everyone, atheist and theist alike — ascribe many of the beliefs of others that we don't agree with to their desires, wishes, hopes, preferences, etc. The fact that we only seem to do this when we disagree with the beliefs — indeed, that we find them "impossible" — is instructive.
This indicates that there is a relationship between belief and desire. The mere existence of "intellectual fashions" points to the fact that there are social influences to the beliefs we have. Factors like the desire for conformity, popularity, and even notoriety can impact what beliefs we hold and how we hold them.
Do we really believe things because we want to believe them, as we often claim about others? No. We believe the best about our relatives not so much because we want to hold those beliefs, but because we want the best to be true about them. We believe the worst about our enemies not because we want to hold those beliefs but because we want the worst to be true about them.
If you think about it, wanting the best or worst to be true about someone is much more plausible than simply wanting to believe something good or bad. This is because our mere beliefs about someone don't necessarily amount to much whereas the truth about someone does. Such desires are very powerful, and although they may be enough to produce beliefs directly, it is more likely that they will aid in the production of beliefs indirectly. This happens, for example, through selective examination of evidence or our choices in what books and magazines we read.
Thus, if we say that someone believes in a god because they want to, that isn't true. Instead, it may be that they want it to be true that a god exists and this desire influences how they approach the evidence for or against the existence of a god.
What this means is that the Queen is not correct that Alice can believe impossible things simply by wanting to believe them. The mere existence of a desire to believe is not in and of itself sufficient to produce an actual belief. Instead, what Alice needs is a desire for the idea to be true — then, perhaps, a belief can be produced.
The problem for the Queen is that Alice probably doesn't care what the Queen's age is. Alice is in the perfect position for skepticism: she can base her belief solely on the evidence at hand. Lacking any evidence, she can simply not bother to believe either that the Queen's statement is either accurate or inaccurate.
Rational Belief
Since it cannot be argued that a rational person simply chooses the best beliefs, how it is that one acquires rational as opposed to irrational beliefs? What do "rational beliefs" look like, anyway? A rational person is one who accepts a belief because it is supported, who rejects a belief when it is not supported, who only believes to the extent that evidence and support allows, and who has doubts about a belief when the support turns out to be less reliable than previously thought.
Notice that I use the word "accept," rather than "chooses." A rational person does not "choose" to believe something simply because evidence points that way. Once a person realizes that a belief is clearly supported by the facts, there is no further step which we could call "choice" that is needed for a person to have the belief.
It is important, however, that the rational person be willing to accept a belief as a rational and logical conclusion from the available information. This may even be necessary when one wishes that the opposite were true about the world because sometimes what we want to be true and what is true isn't the same. We may, for example, want a relative to be truthful but we may have to accept that they are not.
What is also required for rational belief is that a person attempt to assess some of the non-rational, non-evidential things which lead to belief formation. These includes personal preferences, emotions, peer pressure, tradition, intellectual fashion, etc. We will probably never be able to eliminate their influence on us, but just identifying their impact and attempting to take them into account should help us. One way of doing that is to avoid some of the ways in which the non-rational ideas affect beliefs — for example, but trying to read a wider variety of books, not just those which appear to support what you would like to be true.
I think that we can say that the Queen is not going about acquiring beliefs in a rational manner. Why? Because she explicitly advocates choosing beliefs and having beliefs which are impossible. If something is impossible, then it cannot be an accurate description of reality — believing something impossible means, then, that a person has become disconnected from reality.
Unfortunately, this is exactly how some Christian theologians have approached their religion. Tertullian and Kierkegaard are prefect examples of those who have argued that not only is belief in the truth of Christianity a virtue, but that it is even more virtuous precisely because it is impossible for it to be true.
