Yes, very often children in schools are taught to remember it and say it by heart. I heard of a Rabbi who asked his 11 year old son if he knew the 'Tanya' by heart, the child said yes. In return the Rabbi said ok, I'll wake you up in the middle of the night and then you can recite it to me...
The questioner's information is inaccurate and misleading. There is no directive in Judaism that
obligates its adherents to 'memorize' the Torah. Those who engage in regular Torah study (which
is strongly encouraged) naturally find, as a by-product of routine study, that words, phrases, and
verses become known to them, and that they often have in mind a verbatim quote that fits a given
situation. There are certainly individuals or groups who find meaning and value in memorizing
portions of the scriptures. But on the whole, Judaism has no requirement for memorization.
The questioner should take a bit of time for directed reading or Googling, in order to clarify some of
the inaccurate impressions with which he is burdened.
Memorizing Torah passages is not a central lpart of the Jewish tradition. Jews traditionally pray from prayerbooks and read Torah. Literacy is highly prized. However, most Jews can recite key passages from memory, notably the first part of the Shema (Deuteronomy 6:4-9). It is quite likely that all Jewish children in even marginally Jewish families have memorized this by bar mitzvah age. Also, many Jewish boys end up memorizing the Torah portion for their first alyiah (when they are called up to the Torah on their bar mitzvah). This is because reading Torah is traditionally done to a melody (trope) that is tightly connected to the grammar of the text. For those who are really fluent in Hebrew, the trope follows largely from the text and grammar, with a few elements in each passage that must be memorized, but for those who are not fluent, memorizing the entire passage and using the written text as a crutch to spur the memory is a faster track to what, for many less observant boys, is a once-in-a-lifetime event.
Another answer mentions a Jewish child who has memorized Tanya. Tanya is not Torah, but rather, a book by Rabbi Shneur Zalman of Liadi published in 1797. This is an important text to the Chabad Lubovitch movement within Hassidic Jews, and the child who is reported to have memorized it is almost certainly a member of a Lubovitch family.
It depends on how the term is meant. "Torah Jews" can mean more than one thing. If the definition is: Those who live the laws of the Torah (which is the way the phrase "Torah Jews" is usually used), then it is understandably common to reserve that description for observant Jews. It is indeed customary today to call observant Jews "Torah Jews"; so the answer to the question is Yes. The word "Orthodox" is seen by many Torah Jews to be an exonym, i.e. a term applied to them by non-Orthodox, whereas they prefer the term "Torah Jews". If the definition is: Who is Jewish according to the Torah, then Torah Jews would include non-observant Jews, because they don't cease being Jewish. All Jews, regardless of levels of observance are "Torah Jews" since their Jewishness is derived from the Torah's mandates.
They are Jews who keep the Torah.
Orthodox Jews or Torah Jews.
Yes
To honor the Torah and glorify it.
Torah
The Torah.
Orthodox Jews abide by the commands of the Torah and the Oral Torah (Talmud).
No, Torah-observant Jews do not.
The Torah is written ... and read from ... in Hebrew.
The Jews received the Torah from God written in Hebrew, and significant numbers still read and study Torah in the original Hebrew to this day.
The Torah.