Question: Does "one witness is believed in matters of Issurin (what is religiously forbidden/permitted, including, kashrut)" apply even if the witness has a personal interest, such as a store or restaurant? Does it apply to a woman? Must the person be a Yareh Shamayim? How is a mashgiach better than the owner if the business pays him?

Answer: [This is a general, not detailed, answer.] When full testimony is required, i.e., for monetary matters, punishments of beit din, and matters of "family status," two witnesses are required (see Gittin 2b), and they must not have a direct interest in the matter (Rambam Eidut 9:1). Formal testimony is not needed for matters of Issurin, which is the reason that one witness suffices (Chulin 10b).
When one person is enough, a Nogeiah B’eidut, one who is affected by the "testimony" can be used. One example is that a butcher is believed to say that all the steps needed to make meat kosher were done (Rambam, Maachalot Assurot 8:7). We do not suspect him of lying to make money by selling non-kosher food to kosher consumers. The person does need to be under the presumption of reliability on religious matters, which requires him to, first and foremost, be personally observant (ibid.). As a rule, one who eats kosher will not feed non-kosher food to others. Some mainly religious people have serious flaws in their observance of certain areas of Halacha. Then, one might be believed regarding certain areas of Halacha and not others. The rule is that one who violates "light" Aveirot does not automatically lose credibility regarding "heavy" ones; some of the complicated details are found in Shulchan Aruch, Yoreh Deah 119.
Fundamentally, there is no distinction between the reliability of men and women regarding Issurin (see practical distinctions in Rama, Yoreh Deah 127:3). In fact, one source that people can rely on others regarding Issurin is from the Torah’s description of a woman’s counting the days to end her nidda status, regarding which her husband is to trust her (Tosafot, Gittin 2b). Rashi (ad loc.) says that the source is the correct assumption that one can trust the kosher status of food prepared by others, and this applies to both men and to women.
Where did the idea of requiring Hashgachot come from? The Rosh (Chulin 1:24) says that in his time the broad Minhag was not to trust butchers for all of the checking needed but to appoint experts. Mahari Halevi (17) points out that it is not out of fear of purposeful deceit but that some elements may be too complicated for certain butchers who might not admit it.
In some communities, a proprietor who is known to be trustworthy is not required to obtain a formal Hashgacha. However, most communities require some level of rabbinic supervision (the supervision is often looser when the proprietor is known to be trustworthy). Having a mashgiach is "healthy" for the following reasons. 1. Since, as above, even honest people make mistakes, it is worthwhile for someone with training to supervise. He should catch as many mistakes as possible and know how to deal with them after the fact. The mashgiach also has easier access to kashrut experts when needed. 2. One who is new in or passing through town and does not know who is and is not trustworthy can be guided by the certification of known rabbis or organizations. 3. Every once in a while, someone who was assumed to be trustworthy turns out to not be; while Halacha does not demand us to suspect this, extra prudence on matters affecting the public can be positive.
Regarding mashgichim being paid by the people they are supervising, #1 and #2 above are not issues. Regarding #3, the guarantees are indeed lower if the proprietor can pressure the mashgiach financially to not be sufficiently vigilant. However, Halachically, the Hashgacha is still valid. As we have seen, we do not expect trustworthy people to lie about kashrut even if they have a financial interest. However, many organized kashrut organizations pay the mashgiach themselves to reduce the chance of abuse of the system.