You ask four different questions. I shall respond briefly to each, and then add a 5th question based on the neuropsychology of "trust" in order to understand why some people are more trustworthy than others and how "trust molecules" can be cultivated.
1. "Trust" in the Bible refers to being faithful and relying upon the other. The term used is bitachon. One is exhorted to trust in God and not in mortal benefactors (Psalms 115; 118,8-9; 143,3-4). The prophet Jeremiah goes so far as to proclaim, cursed is the person who puts his trust in human beings rather than in God (Jeremiah 17,5). A different construct began emerging in Rabbinic discussions on how to create and maintain just and honest societal conventions. Several Talmudic tractates deal with when and how to trust another person. Thus, for example, Bava Metzia chapters 1-5 deals with a shomer hinam, a trustee who takes care of the property of another person gratis, and chapters 6-7 discuss a shomer sakhar, a paid trustee. Today, in Modern Hebrew, the terms for interpersonal trust are neamanut (faithfulness, loyalty, reliability) and aminut (credibility).
2. Yes, sometimes it is permitted to "betray" trust. Higher values may trump a personal promise. In Jewish ethical thought moral absolutism is a rarity. Many Halachic responsa clarify when to maintain rules and regulations and when other values win out. Thus, for example, trust may be violated for the greater good of saving lives. Similarly, modern ethical theory ranges from approaches such as deontology versus virtue ethics. A deontologist (from the Greek deon, obligation, duty) assumes that the agent should act in accordance with a prevailing moral obligation such as "What is hateful to you, do not do to your friend (the tanna Hillel in Shabbat 31a). Whereas, in virtue ethics, moral reasoning emphasizes an overriding value such as benevolence and the greater good.
3. Trustworthiness of another person assumes character traits such as integrity, truthfulness, honesty, reliability, responsibility, concern and discernment. Conversely, you would hesitate to trust a person exhibiting harmful character traits such as envy, disparagement and excessive egoism.
4. Many mitzvot imply trustworthiness as a value, e.g. "Love thy neighbor as you love yourself", or "do not lie", or "do not covet". Trustworthiness as a virtue ethic is often subsumed under the all-inclusive mitzvah of leading a virtuous life, "Do what is just and good" (Deuteronomy 6, 18). It is implied in moral injunctions such as "You should travel on the path of goodness, and keep the ways of the righteous" (Proverbs 2, 20). Scholars in Orthodox Judaism such as Rabbi Prof. Walter Wurzburger have advocated a Jewish virtue ethic that can be integrated with the Halachic system.
5. Why are some people trustworthy, while others lie, cheat and steal? Paul J. Zak, who directs the Center for Neuroeconomics Studies at Claremont Graduate University, suggests that oxytocin may explain the neuropsychology of trust, empathy and virtue. In our blood and in the brain, oxytocin appears to be the chemical elixir that creates bonds of trust not just in intimate relationships but also in business dealings, politics and society at large. Zak suggests various ways of triggering this "moral molecule" and creating mutual trust.
Zak's book, published May 10, 2012, is worth analyzing from a Jewish perspective. Many practices, norms and rituals embedded in Judaism would seem to foster the triggering of oxytocin in a natural and spontaneous way. This is a byproduct of "social mitzvot" that create supportive caring from birth to death, marriage to divorce. It is worth further investigation to determine if and how communal celebrations, prayer quorums and mandates of virtuous behavior might help engender trust, empathy and kindness on both the individual and societal levels.
 See Eliezer L. Jacobs and Shalom Carmy (eds.), Covenantal Imperatives: Essays by Walter S. Wurzburger on Jewish Law, Thought, and Community, Urim Publishers, 2008. Yitzchak Blau, "The Implications of a Jewish Virtue Ethic", Torah U-Madda Journal 9, 2000, pp. 19-41.
 See Paul J. Zak, The Moral Molecule: The Source of Love and Prosperity, Boston, Dutton Publishers, May, 2012. http://www.moralmolecule.com; video.ted.com/talk/podcast/2011G/None/PaulZak_2011G.mp4
Trust is a fundamental element of Jewish living and Jewish spirituality. In fact, the word for “trust” in Hebrew – emunah – is synonymous with other profoundly important Jewish concepts: “faith” and “truth.” In this light, trust is formed when we can put our faith in another – when we can rely upon the inherent stability and validity of that relationship.
It is noteworthy that the great developmental psychologist, Erik Erikson identifies trust as the first and primary psychological component that forms the basis of our identity. For Erikson, trust or mistrust is formed for the infant in relationship with the mother; by her dependability and her responsiveness to the child, she reflects back to the child his own sense of trust in the world and personal sense of meaning.
Trust in Jewish life begins with truthfulness and fulfilling the commandment: “You shall not bear false witness against your neighbor” (Ex. 20:13). Telling the truth is assuredly a mitzvah of the highest degree. In order to avoid dishonesty, the Talmud wisely advises, “Teach your tongue to say, ‘I don’t know,’ lest you be caught in a lie” (Berakhot 4a).
It makes sense that honesty, truthfulness, and trust are so important in Judaism, for they play a part in everything we do, such as how we speak about others and ourselves, our business dealings, our marriages, and our parenting (stating false promises to children is denounced in the Talmud). It is no wonder that the prayers enveloping the Shema – the central affirmation of Jewish faith – are about emunah, trust and truthfulness.
After all, imagine what life would be like without trust: marriages formed on betrayal and deception; teachers lying to their students; treacherous business dealings; treasonous leaders. Trust is the virtue that anchors our society away from living in precarious paranoia and ruthless social Darwinism.
Of course, however, Judaism is realistic and there may be times when lying might actually contributes to a greater value of peace. That is to say, if lying helps to bring peace or averts injury it is permissible. Classic examples include, telling an unsightly bride on her wedding day that she is gracious and beautiful (B. Talmud, Ketubot 17a) and ascribing the reason for tardiness to synagogue to something other than the truth if the truth was due to sexual relations with your spouse (B. Talmud, Bava Batra 23b). As one 20thCentury rabbinic authority states: “Sometimes it may be wrong to tell the truth about our neighbor… and sometimes it may be necessary to change the details, when the plain truth would injure” (Mikhtav Eliyahu).
As with all Jewish value, trust begins with each of us individually, as it’s told:
Rabbi Simcha Bunim of Przysucha was aksed: “What is the law?” He replied, “It is forbidden to deceive your neighbor.” “And what is going above the letter of the law.” “Not deceiving yourself.”
One of Hebrew’s words for trust, emunah, is also the word for having faith in God, classically Judaism’s most valued form of trust. Judaism has addressed the question of trust between people in many ways; through stories in the Torah, Midrashim, and halakha, Jewish law. Trust is something to be valued, even treasured, but it can be amorphous and difficult to measure, especially over a short span of time. Consequently, the tradition focuses more on describing the qualities and habits that make one trustworthy, than it does on issuing benchmarks for deciding if a person is worthy of one’s trust.
Joseph is the model of a trustworthy person. He comes into Potiphar’s house as a slave but soon his master, “…put him in charge of his household, placing in his hands all that he owned.”(Gen. 39:4) Joseph proved faithful to Potiphar even in the face of his wife’s attempt to seduce him. He would continue to earn people’s trust, while in prison and again as a trusted advisor to Pharaoh. Joseph was even able to earn the trust of his brothers who once despised him for spying on them.
The rabbis are powerfully aware that where money is involved there is always the possibility for violating trust. When Moses calls for gifts to build the ark of the covenant he also gives a public accounting of the wealth that is received. Trust is something that every person, no matter how high his or her office, must earn. The Psalmist alerts us to the reality that our words are our most powerful tools for violating trust, “O Lord, save me from treacherous lips, from a deceitful tongue.” (Ps. 120:2) The fallout from charming words is all about us. Proverbs reminds us that, “An enemy dissembles with his speech but inwardly he harbors deceit.” (Prov. 26:24) We merit trust through our actions. The Talmud describes in detail the kind of discipline needed to merit trust, “A shopkeeper must clean his measures twice a week…and cleanse his scales after every weighing of liquids” (Baba Batra 88b).
However, Jewish tradition also demonstrates an awareness that too literal an interpretation of what constitutes honesty can make one vulnerable to evil, or result in unintentionally hurting others. After repeatedly being cheated by his father-in-law Laban, Jacob realizes he will never come to a fair agreement for his labor and he must simply take his property and flee. This is an extreme action, but the story reminds us that we must be shrewd in the face of evil. We should always seek to establish trust between ourselves and others, but we must also be aware that it can be used as a weapon against us. So too, if someone tells us they are going to kill themselves, we can betray their trust in the interest of pekuach nefesh, saving a life. Jewish tradition has many examples of white lies that are told in the interest of modesty or of not wounding others. Hillel’s disciples exhorted us to sing “O beautiful and graceful bride” to all brides, however they look.
One should trust people of great integrity, but as cases like Bernie Madoff’s teach us, scoundrels have a talent for fooling people. We only have control over cultivating our own character and working at making ourselves worthy of people’s trust, not whether someone else is trustworthy. However we can, with good judgment, and more than a touch of luck, draw close to people who have demonstrated a trustworthy character. Certainly, Jewish teaching reassures us that there have always been people worthy of emunah, even as there have always been scoundrels. Is emunah a mitzvah? Certainly its components are; telling the truth, being honest in business, and cleaving to our covenants with others, but trust itself is a value to be cultivated, often over a lifetime, rather than a defined action that can be legislated.
(Thanks to Eugene Borowitz and Frances Schwartz who’s fine chapter on emunah in Jewish Moral Virtues was particularly helpful in writing this response)