I think these are very good questions.

In my opinion, however, the questions are not complete.   There is no perspective implied in the questions, and I think one is necessary to answer fully.  In other words, are we talking about individually?  Culturally?  Evolutionarily?  (is that even a word, or did I just make up a word?)   The answers will be quite different.

From an evolutionary standpoint, the only goal is for the species to continue.  From that perspective, irrationality is only dangerous when it leads to enough deaths to threaten the survival of the species, or causes some other threat, maybe mass sterility.  Likewise, happiness is not particularly important unless it can be shown that happy people reproduce more successfully, or have stronger offspring, etc...

I'm pretty sure that's not what you're getting at, though.

On a personal level, why rationality?  Well, humans strive towards happiness quite naturally.  All you need to do is watch a baby.  It cries because it's unhappy and wants things to be different than they are.  I know philosophers argue over the exact nature of happiness, and its exact causes, but for the moment, I'm going to take it as a given that people desire to be happy, and that in general, they also recognize happiness, regardless of their ability to define it exactly.

So, if we agree that people try to be happy, we can ask the question, "What do people do to try to be happy?"  The answer, simply, is that they do things they believe will affect their bodies, their surroundings, and their relationships in such a way as to make them happy.  Sometimes they're successful, sometimes not.

What do they use to decide which actions to take?  Logic.  Many times, they have faulty information, and so make poor decisions, which tends to make them more unhappy than people who have good information, and make good decisions.

So, one element in their decision making process is the truth value of their premises.  The other is the logical validity of their thought processes.  If either element is faulty, the end result is less likely to be good (bring happiness).

This is not to say that irrational thinking based on faulty premises can't lead to happiness.  Clearly it can.  However, in the long term, people who are both factually accurate and logically sound will tend to make better decisions than those who are not one or both of those things.

Taking another angle, in order to make decisions, we need data.  Most of this data is not intuitive.  If a person is irrational, all of their data obtained deductively is suspect, and much of it will be incorrect.  So, in this system where we need both good logic and good data, each element props the other one up, so to speak.

Having said all of this, I admit I have not tackled the question of life's value.  Again, this is a matter of perspective.  Evolutionarily, my life has very little value, since I have decided not to reproduce.  Personally, I value my life very highly because I am human, and have a natural desire to live.  Couple this with my natural desire to be happy, and you naturally arrive at the conclusion that my life is quite valuable.  Since I interact with other people, and presumably my presence makes some of them happier than my absence, my life also has value to others.

Is a rational person's life more valuable than an irrational persons?  Depends.  If a rational person becomes a scientist and invents say, a car that runs on water, his life has had great value evolutionarily, since it is likely that this car will slow down the degradation of the planet from overuse of resources.  He has great value to people individually because they'll save money on gas, etc...  A person who is irrational, and spends his whole life promoting a website advocating drinking honey with lemon to cure skin cancer will have very little value to the rest of humankind.  His life will still be personally valuable, and I wouldn't say that it's any more or less valuable than the rational inventor  -- on a personal level.

So, getting back to my original point, perspective is necessary to answer the question of "Why rationality?" or even "Why happiness?"

It's basic evolution.  Humans, like all other animals, have a desire to stay alive.  In the event that a life form evolves to the point where it does not want to live, it will die, and the entire species will die.

Does it matter how much happiness you've had when you die?  No.

Why would it?

It matters while you're alive.  Why should the inevitability of death cause you to want to hurry it along? 

It's just evolution.

Why not indeed. By my reasoning, the answer to your question is axiomatic: Those who see no point in living would have already killed themselves, and not even be here to engage the argument for killing oneself. By merely participating in the argument, you demonstrate that you wish to continue living.

Empathy. Desire. And a survival instinct.

Why not? Seriously. Please give a good reason to give up and end one's life. 

For most people, they have friends, loved ones, and especially children that they care for and love and want them to experience a good and happy life as well, on into the future. So, we do what we can to make the future a better place for humanity. I personally may die, but my legacy can live on in other people. Humanity could theoretically continue living for the end of time. Our descendents could explore other stars and galaxies in the universe. But only if we build that future starting now.

Yes, to be fair,  inkwellmusingsdotcom's question is not about god's existence, but rather, in god's abscence, what motivation have we for being rational.  Nietzsche as an atheist was anti-science.  It is certainly fair to ask whether a rejection of god points us towards nihilism. 

The answer is simply pragmatism. Nihilism is useless, so we choose to believe useful things, i.e. pragmatism. Use the best ideas. Science and rationality are good ideas, they produce useful results, they make useful predictions, so we use them. Nihilism, i.e. the belief in nothing at all, produces nothing.