בשני ובחמישי ובשבת במנחה קורין שלשה אין פוחתים מהם ואין מוסיפים עליהם ואין מפטירין בנביא: הגה ואם היו שני חתנים בב"ה והם ישראלים מותר להוסיף לקרות ד' דלדידהו הוי כי"ט שמותר להוסיף (מרדכי פ' הקורא עומד והגהות מיימוני פ' י"ב מהס"ת) ונראה דה"ה לשני בעלי ברית די"ט שלהם הוא כדלקמן סי' תקנ"ט (ודין שבת וי"ט ע"ל ריש סי' רפ"ב לענין הוספה):

On Monday, Thursday and on Shabbat at mincha [the afternoon service] three read [from the Torah], and we don't subtract from them or add to them, and we don't conclude with [a reading from] the Prophets [i.e., read a haftarah]. Gloss: If there are two grooms in the synagogue and they are "Israelites" [i.e. not Kohanim or Levi-im], it is permitted to add [in order] to read four, because for them it is like a Festival when it is permitted to add (Mordechai on Perek HaKorei Omed [Talmud Megillah, Chapter 3]; Hagahot Maimoni on Chapter 12 of Hilkhot Sefer Torah). And it seems that the same applies for two officiants a Brit [Milah], because it is their "Festival", as it says below in chapter 559 (and [for] the laws of Shabbat and Yom Tov, see below in chapter 282 regarding adding [aliyot]).

The place that we stop [reading from the Torah] on Shabbat morning, from there we [start to] read on [Shabbat] mincha [at the afternoon service], and on Monday, Thursday, and on the following Shabbat [morning]. Gloss: If they missed the public reading of the parasha on one Shabbat, then the following Shabbat they read that parasha [i.e., the missed one] along with the parasha belonging to that [i.e., the following] Shabbat (Or Zarua); and see below, chapter 282 regarding adding [aliyot].

A Kohen reads first from the Torah, and after him, a Levite, and after him, an Israelite.

The widespread custom is that even a Kohen who is unlearned reads before a great [Torah] scholar that is a Israelite [i.e., someone not a Kohen or Levite], as long as the Kohen knows how to read. (However, if he is able to read word for word with the prayer leader, that is sufficient; like it says later in Siman 139 (Abudraham)) For if he doesn't know how to read, how will he bless over the Torah [reading]?

If the Torah scroll is opened and the Kohen is reciting Shema, he is not permitted to interrupt, and they call up a Yisrael in place of him. Gloss: And similarly, on a fast day that follow Pesach and Sukkot, where we read "Vay'chal" - if the Kohen is not fasting, we call a Yisrael [instead]; and it's preferable for the Kohen to go out from the synagogue (Maharik - shoresh 9).

If a Kohen entered the synagogue after a Yisrael began blessing the [main part of the] Torah blessing, one (the Yisrael) does not stop. But [only the introduction of] "Bar'chu", it is not [considered] the beginning [of the blessing], and the Yisrael should stand at the Ark until the Kohen and Levi complete [their aliyot], and then he reads. And if there is no Kohen in the synagogue, the Yisrael reads instead of a Kohen and a Levi is not called up after [the Yisrael].Gloss: But [a Levi] may be called up for the first [aliyah when there is no Kohen present] (Beit Yosef and Mahari"l in the name of R. Yerucham - netiv 2 in the third volume). And when they call [the non-Kohen] up, they say "instead of a Kohen" so that [people] should not err and say that [this person] is a Kohen [Tur].

If there was a Kohen and a Levi in the synagogue and the Kohen read [an aliyah], and [then] thinking that there is no Levi there and started blessing the Torah blessing a second time, we do not stop [the Kohen].

If there is no Levi in the synagogue, the Kohen that was called first blesses a second time and reads in place of a Levi, but not a different Kohen, so that [people] shouldn't say that the first one was invalid.

Similarly, two Levi'im should not be called up, one after the other, so that [people] should not say that one of them is invalid.

The custom is to call a Kohen after a Kohen if a Yisrael was called up between them. And the Chazan should say, when he calls the second Kohen: "Even though he is a Kohen", and we do similarly for a Levi after a Levi. Rem"a: Therefore, it is permissible for them to come up for the maftir aliyah in this way. And if we call them up for the maftir without mentioning their name, there is no need to be concerned people thinking they are invalid since we are not mentioning their name. And there are those who say that we don't call up a Kohen or Levi within the regular seven aliyot, but after the seven have concluded, we may call up a Kohen or a Levi, [Agur in the name of Mahari"o and Mordechi - chapter "Hanizkin", and Beit Yosef in the name of R. Yerucham] and this is the custom in these countries. However, in a case of dire need, one may rely on the first opinion.

There is an opinion that if the chazzan calls up a Kohen or a Levi and they are not there, he should not call another [Kohen or Levi] up by name since [that would raise suspicion that] the first one was invalid. Rather, another [Kohen or Levi] should go up on his own [without being called up]. (And this is our custom. A prayer leader who is a Kohen may call up another Kohen to the Torah.)(Mordechi - chapter "Hakoreh Omeid", and the Agur)

[Regarding] a city of Kohanim - if there is one "Yisrael" among them, that person should read first for the sake of peace. And if there are not enough Yisraelim or if there are no Yisraelim at all, it is permitted to call a Kohen after a Kohen, because there is no cause for suspicion that any [of the Kohanim] are invalid, since everyone knows that there are only Kohanim there. And the same applies to a city of Levi'im.

[Regarding] if a child may read from the Torah in public, [it's discussed] in siman 282. [Regarding] if the only Kohen is blind, or if he lacks expertise [at reading from the Torah], [it's discussed] in siman 139.

[Regarding] people who are being held in prison - we do not bring a Torah to them, even on Rosh Hashana and Yom Kippur. Rem"a: This is specifically at the time of reading, but if they prepare a Torah for him a day or two prior, then it is permitted. [Ohr Zarua, Hagahot Ashiri - first chapter of Tractate Berachot, and Maharam Padua - siman 88]. And if he is an important person, then it is always permitted. [There in Mordechi - end of the first chapter of Tractate Rosh Hashana]