"Buddha never said the Karma and rebirth theory was true"
I don't know if he claimed it to be true or not, but it seems like he spoke about karma and rebirth as being true:
"With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, he directs and inclines it to knowledge of the recollection of past lives (lit: previous homes). He recollects his manifold past lives, i.e., one birth, two births, three births, four, five, ten, twenty, thirty, forty, fifty, one hundred, one thousand, one hundred thousand, many aeons of cosmic contraction, many aeons of cosmic expansion, many aeons of cosmic contraction and expansion, [recollecting], 'There I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure and pain, such the end of my life. Passing away from that state, I re-arose there. There too I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure and pain, such the end of my life. Passing away from that state, I re-arose here.' Thus he recollects his manifold past lives in their modes and details. Just as if a man were to go from his home village to another village, and then from that village to yet another village, and then from that village back to his home village. The thought would occur to him, 'I went from my home village to that village over there. There I stood in such a way, sat in such a way, talked in such a way, and remained silent in such a way. From that village I went to that village over there, and there I stood in such a way, sat in such a way, talked in such a way, and remained silent in such a way. From that village I came back home.' In the same way — with his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability — the monk directs and inclines it to knowledge of the recollection of past lives. He recollects his manifold past lives... in their modes and details."*
"With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, he directs and inclines it to knowledge of the passing away and re-appearance of beings. He sees — by means of the divine eye, purified and surpassing the human — beings passing away and re-appearing, and he discerns how they are inferior and superior, beautiful and ugly, fortunate and unfortunate in accordance with their kamma: 'These beings — who were endowed with bad conduct of body, speech, and mind, who reviled the noble ones, held wrong views and undertook actions under the influence of wrong views — with the break-up of the body, after death, have re-appeared in the plane of deprivation, the bad destination, the lower realms, in hell. But these beings — who were endowed with good conduct of body, speech, and mind, who did not revile the noble ones, who held right views and undertook actions under the influence of right views — with the break-up of the body, after death, have re-appeared in the good destinations, in the heavenly world.' Thus — by means of the divine eye, purified and surpassing the human — he sees beings passing away and re-appearing, and he discerns how they are inferior and superior, beautiful and ugly, fortunate and unfortunate in accordance with their kamma. Just as if there were a tall building in the central square [of a town], and a man with good eyesight standing on top of it were to see people entering a house, leaving it, walking along the street, and sitting in the central square. The thought would occur to him, 'These people are entering a house, leaving it, walking along the streets, and sitting in the central square.' In the same way — with his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability — the monk directs and inclines it to knowledge of the passing away and re-appearance of beings. He sees — by means of the divine eye, purified and surpassing the human — beings passing away and re-appearing, and he discerns how they are inferior and superior, beautiful and ugly, fortunate and unfortunate in accordance with their kamma..."*
source: http://www.accesstoinsight.org/tipitaka/dn/dn.02.0.than.html
You can certainly find more suttas that will direct you in the same direction of thinking.
"it is not important to Buddha's teachings"
Correct. You don't have to believe in karma and rebirth to achieve complete liberation from suffering.
The main teaching of the Buddha is to do wholesome actions and meditate to find out the cessation to suffering.
"Siddhartha Gautama's contemporaries believed in Karma like we believe in the Big Bang theory and evolution. Neither of these theories can be proved or refuted with an affordable effort (in order to prove evolution for yourself, you need substantial knowledge of biology plus systemic thinking skills -- both of which are hard to acquire)."
Karma exists. Rebirth exists.
Karma does not exist. Rebirth does not exist.
Karma neither exists nor not exists. Rebirth neither exists nor not exists.
All three statements above are true. All of this can be known and seen.
None of the above statements will ever be possible to be 100 % proven with evidence, only with experience, which, unfortunately, is not easy to pass to others.
"Also, neither of these theories affects the main purpose of Buddhist practice -- to reduce suffering. If tomorrow someone proved the Big Bang theory to be definitely true (or definitely wrong), it wouldn't make most of the people happier/unhappier than before."
Your statement is correct. Belief in kamma and rebirth doesn't ALWAYS reduce suffering. Example is an individual who believes in kamma and rebirth very much, but still does such unwholesome actions to end up in a place of deprivation.
Your statement is not correct. Belief in kamma and rebirth reduce suffering. Example is an individual who believes in kamma and rebirth very much, thus he abstains from doing unwholesome actions and ends up in a place of happiness.
Both statements above are true. All of this can be known and seen.
"If someone does not believe in the Karma theory, they will focus on making the best of this life (which may be the only one we get)."
Again your statement is correct, but it can be incorrect for another.
All of this can be known and seen.
You are looking at a wall that has no visible colors, but you are seeing colors on the wall.
You are searching for answers to questions where all answers are true, untrue, true and untrue, neither true nor untrue.
Either one of these views: true, untrue, true and untrue, neither true nor untrue are conditioned by ignorance.
Knowing that all views are true, untrue, true and untrue, neither true nor untrue is cessation of ignorance. Is cessation of clinging and craving. Is complete cessation of suffering.
"Is the Karma theory an essential component of Buddha's teaching or not?"
Buddhist practice is a means to end suffering.
Karma theory is not needed to achieve liberation from suffering.
Don't bother yourself with questions about karma and/or rebirth. You don't need these answers to achieve the end of suffering.
To achieve the end of suffering you need to meditate and look closely at reality as it really is.
Once you will see reality as it really is in its entirety, you will know the answers to all questions.
EDIT:
Think again before marking my answer as the correct answer:
If you know that reality is that there is no kamma nor rebirth, you are thinking wrongly. If that is the case, you don't know the cause of kamma and rebirth, thus you're subjecting yourself to future suffering.
If you know that reality is that there is kamma and rebirth, you are again thinking wrongly. If that is the case, you don't know the path to the cessation of kamma and rebirth, thus you're subjecting yourself to future suffering.
If you know that reality is that there is neither kamma nor rebirth, and that when there is kamma and rebirth they always arise with a cause, you are thinking correctly. If that is the case, you maybe know the cause of kamma and rebirth and the path leading to their cessation, thus there is a chance you're completely liberated from suffering.
If it is not the last, I again say to you: To achieve the end of suffering you need to meditate and look closely at reality as it really is.