Yes
It depends on the parent. Depending on the parents' level of Torah-observance, the children will have a greater or lesser secular education, and a greater or lesser Torah-education. Very many Jewish homes teach at least some basic observances and beliefs.
Education is and has always been very important in the Jewish community.Every parent taught the children the basic laws, observances and customs of Judaism. From around the age of 5, Jewish boys attended cheider, where they learned Torah and Talmud.
The central place for education in Jesus day was the synagogue. Boys ages 5-18 were taught by Rabbis. The Old Testament was the source of material studied, especially the Torah, the first five books of the Bible. Children were required to memorize the Torah!
By keeping the laws of the Torah and by Jewish education.
Education has always been paramount to Jews. The Torah commands us to learn and teach, and especially to teach our own children. In Talmud Torah, in Yeshiva, and also when at home, children are taught Torah and Torah-subjects. Illiteracy has always been extremely low among Jews, even in eras when the surrounding nations were largely illiterate. Torah-learning is intensive, both requiring as well as inculcating high intelligence. As soon as babies begin to understand the spoken word, Jewish parents start accustoming them to verses of Torah and whichever laws and customs they can comprehend. Excellence in Torah is always encouraged and rewarded.
Prayers and Torah-reading.
Children have the job of learning the torah, so later on they can tell teach their children. So the Jewish people will never die out.
If your maternal grandmother is Jewish, her children are Jewish, including your mother, your mother's siblings, and all of your mother's children. If your paternal grandmother is Jewish and your maternal grandmother isn't, then you're not Jewish according to Torah-law.
That depends when. In ancient times, many strove to reach prophecy. Then and since then, Torah-scholarliness has always been a Jewish goal. That, and keeping all of God's commands such as Shabbat-observance, a kosher home, Torah-education for the children, etc.
The Torah defines the Jewish religion.
The objective of universal education became an early goal in the Jewish culture exactly because the Torah is to be accessible to everyone, beginning from roughly age five or six. It's the father's job to teach his children, or to arrange and provide for their education, but the father by no means has any exclusive responsibility as the reader or studier in the family.
Simchat Torah.