Why are the 16 Unwise Reflections ("shall I exist in future" etc.) considered unwise?

In regards to the unanswerable questions, here is the original Sutta that lays them out:

"There are these four unconjecturables that are not to be conjectured about, that would bring madness & vexation to anyone who conjectured about them. Which four?

"The Buddha-range of the Buddhas[1] is an unconjecturable that is not to be conjectured about, that would bring madness & vexation to anyone who conjectured about it.

"The jhana-range of a person in jhana...[2]

"The [precise working out of the] results of kamma...

"Conjecture about [the origin, etc., of] the world is an unconjecturable that is not to be conjectured about, that would bring madness & vexation to anyone who conjectured about it.

"These are the four unconjecturables that are not to be conjectured about, that would bring madness & vexation to anyone who conjectured about them."

http://www.accesstoinsight.org/tipitaka/an/an04/an04.077.than.html

By saying that they lead only to "madness & vexation" I think it is clear that this text is saying that these questions don't really have any answers.

Now let's move on to the part from the Sabbasava Sutta:

"And what are the ideas unfit for attention that he attends to? Whatever ideas such that, when he attends to them, the unarisen fermentation of sensuality arises in him, and the arisen fermentation of sensuality increases; the unarisen fermentation of becoming arises in him, and arisen fermentation of becoming increases; the unarisen fermentation of ignorance arises in him, and the arisen fermentation of ignorance increases. These are the ideas unfit for attention that he attends to.

...

This is how he attends inappropriately: 'Was I in the past? Was I not in the past? What was I in the past? How was I in the past? Having been what, what was I in the past? Shall I be in the future? Shall I not be in the future? What shall I be in the future? How shall I be in the future? Having been what, what shall I be in the future?' Or else he is inwardly perplexed about the immediate present: 'Am I? Am I not? What am I? How am I? Where has this being come from? Where is it bound?'

"As he attends inappropriately in this way, one of six kinds of view arises in him: The view I have a self arises in him as true & established, or the view I have no self... or the view It is precisely by means of self that I perceive self... or the view It is precisely by means of self that I perceive not-self... or the view It is precisely by means of not-self that I perceive self arises in him as true & established, or else he has a view like this: This very self of mine — the knower that is sensitive here & there to the ripening of good & bad actions — is the self of mine that is constant, everlasting, eternal, not subject to change, and will stay just as it is for eternity. This is called a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. Bound by a fetter of views, the uninstructed run-of-the-mill person is not freed from birth, aging, & death, from sorrow, lamentation, pain, distress, & despair. He is not freed, I tell you, from suffering & stress.

http://www.accesstoinsight.org/tipitaka/mn/mn.002.than.html

From context I think it's quite clear that the Sabbasava Sutta is talking specifically about focusing on views about a self, which according to Buddhism is just a concept.

You ask the question:

Does The Buddha (1) recognise the possibility of finding the truth for these questions, or (2) is it impossible to find the truth, or (3) is it detrimental for us to know the truth?

I think that the right answer is that none of these questions actually have answers because they presuppose things that aren't true or a general formulation might not be possible (for example, the results of Karma are unanswerable because in order to give an answer for a specific case you would need to be able to see all of that person's Karma, which goes back infinitely far into the past. Only a fully enlightened Buddha is able to do that, and even then, that is for a single specific case. Because even a single case is infinitely complex, it is impossible to come up with a "unified theory of Karma" that can give a complete explanation.)

while science and Buddhism share some similar methodology, their ultimate goals aren't the same. Science aims to increase knowledge and understanding of how the natural world works while Buddhism aims to eradicate all defilements of the body, speech, and mind and to put an end to suffering. Further details are available from Ven. Thanissaro's great essay called "Questions of Skill" ( http://www.accesstoinsight.org/lib/authors/thanissaro/questions.html )

Is there a better (than Wikipedia) explanation why we should not reflect on these questions?

Perhaps a better explanation is the Sabbasava Sutta itself.

The 'unwise reflections' are diverse (16 if you count them) opinions about the 'self':

"This is how he attends inappropriately: 'Was I in the past? Was I not in the past? What was I in the past? How was I in the past? Having been what, what was I in the past? Shall I be in the future? Shall I not be in the future? What shall I be in the future? How shall I be in the future? Having been what, what shall I be in the future?' Or else he is inwardly perplexed about the immediate present: 'Am I? Am I not? What am I? How am I? Where has this being come from? Where is it bound?'

This causes views or opinions about the self:

"As he attends inappropriately in this way, one of six kinds of view arises in him: The view I have a self arises in him as true & established, or the view I have no self... or [etc.]

These views aren't helpful:

This is called a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. Bound by a fetter of views, the uninstructed run-of-the-mill person is not freed from birth, aging, & death, from sorrow, lamentation, pain, distress, & despair. He is not freed, I tell you, from suffering & stress.

The are other views to be attended to instead:

"He attends appropriately, This is stress... This is the origination of stress... This is the cessation of stress... This is the way leading to the cessation of stress. As he attends appropriately in this way, three fetters are abandoned in him: identity-view, doubt, and grasping at precepts & practices. These are called the fermentations to be abandoned by seeing.

(i.e. the 'four noble truths').

And the rest of the sutta describes dwelling with restraint; using things moderately; tolerating various things; abandoning various things; destroying (not tolerating) various evil thoughts; developing various virtues.

When the practitioner is successful, then he or she will have abandoned 'fermentations' by the various things above (restraint, use, tolerating, abandoning, destroying, and developing), and will have achieved the goal:

He has severed craving, thrown off the fetters, and — through the right penetration of conceit — has made an end of suffering & stress."

Even though what I gather generally is that Science and Buddhism are consistent in approach, is Science's relentless search for answers bad for people as well?

Buddhism doesn't try to answer everything.

Science tries to explain why the gravity on the moon isn't as strong as here; why chemicals react with some chemicals but not others; how photosynthesis produces oxygen; how the universe began and whether it will end; etc.

Buddhism is more like a single scientific procedure, a (repeatable) experiment, or a recipe: i.e. Buddhism is a way, teaches a way, to end suffering.

Within that context, and assuming that you want to end suffering, questions or view which don't lead to the end of suffering (which includes 'attending to' those views about the self) are called "unwise".

I don't mean to say that Science itself, or the pursuit of science, is wrong: it may be in the category of Right Livelihood.