1) What are the virtues of learning to wait or having patience, סבלנות?
2) Is there anywhere in the Torah that shows because of lacking patience or the ability to wait, caused tragedies throughout Jewish history?
the first teaching in Pirkei Avot is "to be patient in judgment", meaning to weigh things and not make hasty decisions. Likewise, in the Mesilat Yesharim the first step on the ladder of ascent after torah study is watchfulness which is to "not do any act without first weighing it on the scales of understanding". This is the first and most important foundation of everything after torah, i.e. getting the right understanding.
The first tragedy, namely, Adam's eating from the forbidden fruit also involved lack of patience. According to the opinion that it was a grape, had he waited until the Sabbath, he would have used it to make Kidush and drank from it.
Patience, in the sense of trusting in God and letting things take their course, is one of the core teachings of the Torah. Impatience is caused by a false feeling of control over our destiny, and as we learn to relinquish that control to God our patience grows, together with our Emunah. Also, growth takes time, and without patience we do not give ourselves the time we need to grow and mature.
While the Torah does not explicitly instruct us "be patient", patience is a common theme in many of the stories of the Torah. Here are a few examples:
1) Hebrew language have many words for "patience" corresponding to the varied meanings of this concept. According to the sages (Akeidas Yitzchok 27:13) this is the most important personal trait to be pursued. There are lots of examples showing the virtue of that; I think is worth mentioning two in particular: restraining anger and avoid idolatry.
2) One midrash says that if Israel had waited for Moses and did not had built the Golden calf, there would have been no exile.
The Nach frequently teaches us to pratice patience with adages such as Koheles (7:8) which says "the patient in spirit is better than the proud in spirit". Mishlei says: “better the long patience than a hero, and someone who is in control of his spirit, than the conqueror of a city” and 14:29: “Patience results in much understanding; Impatience gets folly as its portion.” The prophet Micah show us that he suffers many challenging conditions and yet endures. At some point he says "I will wait for Hashem who saves me" (7:7).
The patience of Israel is expressed in the 12th article of Maimonides' Principles: "I believe with perfect faith in the coming of the Mashiach, and though he may tarry, none the less will I patiently hope every day until he does come."