THE TORAH READINGS ON MONDAYS AND THURSDAYS
Why are they referred to as 'Dorshei Reshumos'?
Rashi: Reshumos are Pesukim, meaning 'those who expound Pesukim'.
Ben Yehoyada: Those who expound, using the 13 Midos with which the Torah is expounded. Reshumos refers to matters which are not written explicitly, but must be expanded and expounded through these Midos.
Iyun Yaakov: Dorshei Reshumos refers to mere allusions, rather than full Drashos.
How is the Torah comparable to water?
Vilna Gaon: Torah is compared to water, as it purifies a person. Sometimes a person who begins to learn might think that he has become purified and his Yetzer Hara has left him. The Pasuk therefore says 'all waters' - just as 40 Se'ah of water is needed to purify a person, so that he can immerse his entire body; so too with the study of Torah. In order that the Torah should purify him and crush his Yetzer Hara, he must be fully immersed in it.
What is the significance of three days?
Maharsha: The Torah is described as a Tree of life 'Shemachazikim Bah' - 'for those that hold onto it'. Anything done three times creates a Chazakah. The opposite phenomenon occurs when doing the opposite, meaning not holding onto the Torah for three days. Chazal therefore enacted the public reading from the Torah with breaks of less than three days, so that 'one who heard the old, will be able to hear the new' (Berachos 40a).
Toras Chaim: Every day, the Yetzer ha'Ra, the evil inclination, renews his attempts to overcome a person. Torah is the antidote to it. After one or two days, a person can still overcome the Yetzer ha'Ra with Torah study. Once there have been three days of attack by the Yetzer HaRah without the defense of Torah study, it becomes like a three-ply cord that is not easily broken, as three times makes a Chazakah.
READINGS EVERY THREE DAYS
Why did the Gemara use the word 'sleep' in the phrase 'they should never sleep three days without Torah?
Maayan Mayim: Even if the Torah is read on Monday and Thursday, there could be a situation where three days have passed, such as if the Torah was read later in the morning on Thursday than on Monday. The word 'sleep' shows that such a case would not be a problem. Reading in the daytime protects the person from the Satan the following night, whatever time during the day that it was read.
THREE MEN, TEN VERSES AND THE TEN BATLANIM
Who were these ten idle people and what was their task?
Yerushalmi Megilah: According to the Chachamim, they were supported by the community to be present in the Shul when the prayers begin. (Hash-m, kivayachol, becomes angry when He comes to the Shul at the beginning of the prayers and there is no Minyan present.) According to R' Yehudah, the community does not need to support them as there will anyway be ten people present in the Shul, who came there to learn.
Teshuvos HaRosh: They are there, so that there will always be a Minyan of people who answer the Berachos of the Shliach Tzibur with Kavana.
Rashba: When there are not ten fixed people coming, even others stop coming, as they say, "Why should I come? I will not find ten people there for a Minyan!"
Ha'Maspik: It is fitting to choose modest people who separate themselves from the pleasures of this world, who spend all of their time in the Shul, for Torah readings and serving Hash-m. They serve as examples to the community for anyone who is searching for Hash-m.

