I have been told that there are conditions that a man must fulfill before he is allowed to read the Torah in public.
If so, what is the halacha? What are the disqualifications?
[EDITED] Example: I have been told that someone who is not "Shomer Shabbat", or a "Kohan" who has married a Giur may not read from the Torah. These are the sort of disqualifications I mean. I am sorry that I am bad at expressing myself.
I will try to add to this as I find more info, so this is a partial answer.
It is a given (it should be, at least) that the reader must be able to pronounce Hebrew pretty much flawlessly. This is not to say that, inevitably, even an expert reader wouldn't make occasional mistakes. But, someone who has trouble pronouncing most Hebrew words or can't read Hebrew without vowels should not be the reader. (See my comment above re what often happens in Reform and Conservative shuls, and, sadly, a number of Orthodox ones, as well. major problem!)
See this article regarding giving aliyot to non-Shomer Shabbat people. Rav Feinstein ruled that if the person is an avaryan - e.g. he is not Shomer Shabat because he feels he must go to work, you can give him an aliyah and thus, have him read (See general rule, explained, below.)) However, those that publicly violate Shabbat may not be given an aliyah, and thus, may not read. Orchos Rabbeinu (p. 122), in the name of the Steipler Rebbe does not make this distinction and says that any non-Shomer Shabbat person may receive an aliyah.
Certain disabilities disqualify a person from reading the Torah in public. I should mention that in understanding public Torah reading, the general rule is that we assume that the original obligation is for each oleh to read his own section. You'll find this to be the general guideline in O.C. It's only later that the Shat"z became the shul Torah reader, and later in time, the Shat"z's role was "reduced" to just tefilla and shuls had designated Torah readers. Thus, when reading the halachot, you can assume that if the oleh is disqualified to receive an aliya because he can't read for his own aliyah, it's a given that he can't be a ba'al Kri'ah reading an entire parsha.
Having said this, see pages 171 - 178 in this Google book. There is a debate as to whether the blind can read from the Torah. The debate stems on a combo of the rule of "written words of Torah should not be recited by heart". Does this rule apply to individual / private Torah reading or public reading? Also see O.C. 139:2 - debate between Mechaber and Mahari"l
Most opinions say that a mute cannot read because he can't audibly say the words. However, it seems that many opinions allow someone who uses a voice processor to be able to read.
The deaf pose an interesting question. There's a question as to whether sign language can be sued as a means of Torah reading. I'll post a MY link on this aspect, later, as I asked this question a while ago. Most current rabbinical opinions state that the standard halachic cherish definition mentioned in the Talmud does not apply to most deaf people today as most can speak well. Numerous others use hearing aids and cochlear implants (which BTW are considered more halachically acceptable than aids regarding halachic hearing questions.) Thus, most opinions are that the deaf can read, but, you may find some opinions that disqualify them.
