Do you become unhappy when happiness disappears?

Happiness is used to mention Nirvanic bliss ("Vimukthi suka").
According to that the other mind states are Unhappy(Dukka).
Because the roots of unhappiness(" greed, hate, ignorance) are still in the background of mind. So there can be blissful overjoyed states to sorrowful miserable states.
In between there can be deep sleep states. But when we awake, we are in the same (dull/joyful) world. It can be change in between any time. But if we have a 'State' that not change/varying for worldly things('8 world dhamma'). It is called the real Liberation/ Happiness.

There are many states of mind that can be categorized either as happy or unhappy. This categorization is individual. This categorization arises because of lack of knowledge about our-self and the reality outside our-self:

For a lay person:

Happiness is satisfying desires.

Unhappiness is not being able to satisfy desires.

For a lay person, he becomes unhappy when happiness disappears.

When happiness disappears, desires are not satisfied and a lay person becomes unhappy. He has 3 choices:

1. Satisfies his desires -> becomes happy
2. Doesn't satisfy his desires -> is unhappy
3. Knows his unhappiness, the cause of his unhappiness and removes the cause of his unhappiness -> becomes enlightened.

For an enlightened person:

There is no happiness.

There is no unhappiness.

By abandoning both happiness and unhappiness, a person becomes liberated. Liberated is beyond happiness and unhappiness. It is neither happiness, nor unhappiness. It is the ultimate. The all and beyond all.

An enlightened person cannot become unhappy. He knows unhappiness leads to a bad future. Knowing, he abandons unhappiness.

An enlightened person can become happy. He knows happiness which leads to a bad future and happiness which leads to a good future. Knowing, he abandons happiness which leads to a bad future and accepts happiness which leads to a good future.

You are very correct to point out the false dichotomy of happiness and unhappiness. If you are not unhappy then surely you are happy? Perhaps it's the way that language works that makes us think in black and white terms. Or maybe it's the way that our minds work that makes language black and white. However as Ben Goldacre says - things are a bit more complicated than that. There are many states that equate to happiness and unhappiness and in between states too.

If you consider the Dyana factors then there are states accessible that normally people might equate to happiness but in fact are subtle (and not so subtle) manifestations of various mental processes. For instance rapture (pīti) and bliss (sukha) could both be said to be related to happiness but are different. Then after this there is equanimity (upekkhā) which could be though of as happiness but could be thought of as a stage in between.

But it isn't just Buddhists that recognise the subtley in happiness. Aristole spoke of happiness as flourishing (eudaimonia) but this wasn't something that could be said to occur moment by moment. I've heard it said that it someone could only have been judged to have (or have not) eudaimonia at the end of their life. So in a way that is happiness - but not as we know it.

If we can perhaps indulge in some dichotomies in the Pali language. It have been taught that a antonymn of dukkha (suffering) is sukha (bliss). Which seems straightforward. However if we consider what the four noble truths tell us then the end of suffering is Nirvana. Is Nirvana then the inbetween state?

When you are hungry and you take a first bite of your favorite meal, the happiness in the aspect of a pleasant feeling that you experience is not actual happiness, it is not in the nature of happiness: it is contaminated happiness. That pleasant feeling you experience is merely a relief. You experience the first bite as [contaminated] happiness only because it relieves you from a greater suffering (i.e. hunger), not because it is in the nature of happiness. It is actually in the nature of suffering; it is suffering of change (a quiet subtle form of suffering).

As Tsongkhapa puts it in the Middle-Length Lam Rim:

At present most of the happy feelings that increase our attachment are minds of happiness that arise with respect to a relief of suffering.

If you eat when you are not hungry, there is no such relief, no such pleasant feeling, no such experience of "happiness in the aspect of a pleasant feeling", and it is experienced as suffering of suffering (a quiet gross form of suffering). You are not mislead to conceive of the object as being in the nature of happiness.

There are three types of equanimity. The first of the three is 'equanimity feeling', which his neutral feelings. Furthermore, when pleasant feeling arise, you wish to associate further with it. When suffering arises, you wish to be separated from it. Tsongkhapa writes further:

That which is neither pleasant nor suffering is that which, when it arises, neither of the two wishes occurs.

You can think of it this way: you enter a crowded room, you look at people and you immediately have “I like this one... I like that one... I don't like this one... and so forth” in dependence upon pleasant and unpleasant feelings. There also will be plenty of people you will not have really noticed, you are indifferent to, your attention wasn't drawn towards, because of neutral feelings.

Another more abstract examples of arising of neutral feeling (which is other than neither-pleasant-nor-unpleasant feeling arising in the 4th concentration of the form realm) is: the conception of the body as permanent , that produces neutral feeling and increases ignorance. Tsongkhapa also adds in the Lam Rim Chen Mo that:

In the case of [these] neutral feelings, ignorance ceases when you see that they are impermanent, permanent, exhaustible, and perishable in nature.